I think a roofer might be the way to go to repair the leaking roof but then he is going to just be fixing the symptoms of your problem. It sounds like you have a massive ice dam problem going on your roof. That is caused by either not enough attic insulation AND/OR warm air from the rooms below getting around the attic insulation. All the warm air getting into the attic (attics should be as cold as possible) makes the snow on your roof melt which then forms ice that expands and makes holes all over. When it then melts you get water coming down into the holes.
Two things: Go outside if you can and look at your roof. If theres still alot of snow on the ground and some on your roof, the areas where you see snowmelt and no snow are the areas where you are lacking enough insulation and or air sealing. Next I would get in the attic and measure the depth and what kind of insulation you have (a really cold climate should have 12 inches) also check all penetrations into the attic like plumbing vents duct chases chimney chases to see if they are sealed. I bet they arent. Where are you located? Do all this in addition to repairing the holes in the roof and your problem should go away.
Good luck.

By doing this I meant repairing the holes with a roofer or handyman and also adding more insulation and sealing all penetrations into the attics with caulk or sprayfoam. Try to air seal in the attic first as caulk and sprayfoam are cheap from lowes or home depot. Then try to add more insulation if you can afford it. Fiberglass is cheap and easy but blown cellulose is way better at sealing and can also be a diy thing for attics. It really costs about the same but is a 2 person job. Read some threads here on adding attic insulation and google cellulose insulation.

OK, what you have going on is known as an ice dam. A very common thing in the Northeast, especially in years where there is early snow when the sun is so low in the horizon, and the temps are very cold as they have been recently.

The popular long range fix is roof work and added insulation, etc, but you can make a big difference right away by getting a roof rake and removing the snow after every snow fall.

Get a roof rake with a lot of extensions and you will be able to reach a roof above a two-story-with-full-attic building. The good news is that you don't have to rake the entire roof, just the sections where you are getting ice dams. When using the rake be VERY CAREFUL around overhead electrical lines. It takes a bit of practice to get used using the roof rake, and for years I left it up to husband. Last winter he was unable to do so I tried it on my own and though it took some effort, it is do-able, even by a very middle-aged woman. It especially helps if there are adjacent roofs (wings, additions) you can perch on while raking so you're not working at such a high angle. I don't do it from a ladder, though, even though I am quite at home on them. I'm either standing on the ground or another roof.

Now about ice dams: first of all be aware that there are two things called ice dams. One type is the little metal brackets that people install on the roof to keep masses of snow from falling as a thunderous sheet over doorways. That's not what were talking about here.

For this discussion an ice dam is a place where you have melted snow that freezes on the roof which eventually builds up and blocks the downward flow of other melting snow. This backs the water up under your shingles allowing it to run down the exterior sheathing or interior walls and ceilings, sometimes for great, perplexing, distances.

The most commonly cited cause of the ice dams are houses without sufficient insulation to keep the heat from rising through the exterior walls to the roof, melting the bottom layer of snow, which then refreezes when we have really cold periods, crating the ice dam. The conventional cure involves added insulation, blocking intra-wall air-channels, sometimes removing and relaying roofing with Grace water shield underneath it, etc. Unless you actually have a frank hole in your roof, with uncontrollable cascades of leaking in, I would try to put all that off for now. (Though you will have tons of contractors scaring the pants off you if you call them out to look at it.)

Let me tell you about my experiences with ice dams. I live in the NE in a very old house with no insulation in the walls. And none to very little in the attic. Some years we have had ice dams, but none since we started removing the snow from the roof after every storm in the early season. (By March it becomes less urgent because there is more sun to melt all the snow off promptly, but right now it's a problem.)

One thing I have noted about these old houses is that when they were built they didn't seem to have such a big problem with ice dams (and certainly no lightweight plastic or aluminum roof rakes to attack it with). It stands to reason that they would have altered or modified building techniques if ice dams were such a persistent problem as they are today. I think it is related to how we heat our homes, particularly because we now heat with radiators along the outside walls, warming that air column straight up to the eaves where ice dams start. And we keep our houses much warmer than what was expected before. So in some ways, I think we have created a problem where there wasn't one to begin with.

However, the customarily recommended long term cure is to fill the wall cavities with insulation blocking the rise of heat along the eave walls. (Though adding insulation has its own, different, issues.) In addition one can remove and relay the roof shingles over a good water shield. One can use electric ice melting cables along the eaves. (I am wary of these as there have been some reports of fires connected to their use.)

Or one can rake, which is what I do. Plus I maintain a careful watch on any spot where I have ever experienced any leaking and am prepared to interupt, dry or, just collect any flowing water.

Since you now own an older house, you may feel overwhelmed with the new things you will need to learn about the care of it. The good news is that many, if not most, issues respond to lower-tech solutions. But it does take some time and effort to get in the old-house groove, which is very different from newer houses and much conventional wisdom is dead wrong when applied to older houses. The problem, at first, is figuring out which is which. But you have arrived, by chance, at a good resource. I'll attach a link to another excellent forum below.

I have often thought it would be a wonderful invention if one could figure out how to apply a pnneumatic de-icing boot to the roof similar to what is used on the wings of aircraft. Have the right conditions, then press the switch, and the boot inflates and all the ice is just popped off, autom atically. (The good thing is that when raking the roof, you will get very warm from the effort, and it certainly qualifies for your daily stint of aerobic conditioning.)

Molly~

Here is a link that might be useful: Old house web discussion forum

We had a month of unusually heavy snow for this part of the country last year, which resulted in layers of snow on the roof of the 1920's addition to the house and ice dams over the 2 valleys that were poorly flashed when this addition was put on the house. We had 2 ice dams and interior leaking over a period of time because the sun never hit the snow/ice on those parts of the roof directly and it just wouldn't melt completely off!
So...we did get a ladder out and we chopped away at the snow and ice over several days time. It was hard work, and we were not very efficient at it because we couldn't get on the roof (a slippery metal one). Eventually the leaking stopped, the snow/ice either melted or we had chopped it away, and all seemed to have passed...until June, when the parts of the stucco ceilings below the leak crashed to the ground suddenly one day! Fortunately no one was under them when it happened! So you do want to be careful and examine the areas where the leaking is occurring to make sure that the ceiling isn't so badly damaged from water that it is in danger of collapsing altogether.
We have since redone the ceiling, reinsulated the attic, and have found all the little holes in the roof where the water from the melting ice dam had backed up and leaked in to the house. So far, so good. We've already had a couple snows with no ice dam leaking inside...but we do check the roof and attic periodically to make sure we're really seeing what's going on up there.