Many people confuse moles and voles and while they are both rodents they are different. A vole is a field mouse that eats vegetation. A mole lives under ground, although it may on rare occassion surface and feeds entirely on insects and earthworms. Some people think moles eat grubs and they do but earthworms are the prime food source for them.
Each mole needs a fairly large chunk of territory to exist and will fight to the death to keep any other moles off that territory, so while many people swear they have several moles they will only have one, very active, mole. Unfortunately, getting your soil into a good healthy condition means you will also have the prime food source for the mole, earthworms.
To get your soil into a good healthy condition you need to know some things about the soil you have.
How much organic matter does your soil have?
How well does that soil drain?
How well does that soil retain moisture?
What is the tilth of that soil?
What does that soil smell like?
What kind of life does that soil have?
The presence of moles does not necessarily indicate a lot of soil life, just that moles are present and searching for food.
"Conventional" garden theory is that you dump some stuff with large amounts of Nitrogen, Phosphorus, and Potash on your soil and that is all it needs and some "organic" people simply substitute other, not synthetic, sources of the same stuff which usually does not get a good healthy soil. Work on the soil anf the grass will grow in.

I did some reading on moles/voles when I realized I would sink when walking across the lawn. While not 100 percent, I tend to think mole, but I am certainly no expert. We do have some earthworms, but not as many as I would like. The tunnels that critter makes are seen above ground if that helps.

How much organic matter does your soil have?

Not near enough, but since we moved here last summer, I have mulch mowed with the exception of spring and fall weed mowing. Where I was not seeing any earthworms before, I am seeing some now.
That is one reason I am thinking about adding compost despite all of the weeds.

How well does that soil drain?

It's a sandy soil and drains well. A bit of a learning curve after having heavy clay soil. However, I do deep and infrequent watering during the hot months to encourage a deeper root system with the grass I do have, even with it drying out quicker.

How well does that soil retain moisture?
Drys out pretty quickly. I learned very quickly that heavy mulch is a must in the flower beds.

What is the tilth of that soil?
Gritty. Super easy to dig in, but has a quite a few sandstone type rocks mixed in. The east side of my backyard is solid rock about and inch down all the way across. There is also a boulder in my backyard. We live less than a mile from a lake as well.

What does that soil smell like?

Overall not alot of smell to it, but in some areas where I have been letting organic matter decay, it is starting to get that earthy smell.

What kind of life does that soil have?

Well, probably not enough. I found the previous owner's stash of lawn chems they left behind...so it has just been coming back to life since we moved in. I noticed after my first season here that we started to have alot more lizzards...alot of them, but on the other hand, I also discovered we were over run with a bug I never got a solid ID on. I had posted on one of the other forums about that last spring..there was a lot of speculation they were stinkbug in the adolescent age, but apparently holds a close approximation to a good bug as well.

These simple soil tests can be of some help, but you do need to get more organic matter into your soil.
1) Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains� too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.

I can sympathize. The weed issue is the only thing preventing me from being 100% organic. I don't have the issue now, because of a healthy thick lawn and organic lawn care regime. But...

If I were to move to a new place and the yard was invaded with weeds and needed a huge overhaul, that is when I will use chemical herbicides. I wish I could be a purist, but there just aren't any good organic herbicides that will leave grass untouched.

If I were you, I would simply use chemicals as sparingly as possible to get things to a manageable state and then maintain and improve organically from there.

This is my first spring in a new house in George West, TX. The previous owner had a 30x40 garden plot that is now 100% sticker burrs. It is not nice. I'm going to encourage the St Augustine to move in and take over. To do that I am going to water only at the edges where the St Augustine is with a soaker hose set on a very slow drip. A slow drip means 1 cup per minute out of the faucet. And I will fertilize the edges heavily with corn gluten meal and alfalfa pellets. As the grass moves into the middle, I'll move my watering and fertilizer in as well.