OMG!

I enjoy antique furniture, but pieces do need to be practical. Drawers over my head ain't practical.

What did the appraisal say?

Good point, Fori....I wonder how those drawers were used? Library steps?

Very nice indeed. Looks to be all original, chest-on-chest right? I know that there are many drawer arrangements, have never seen one that did not have graduated drawers on top. I assume there are partitions inbetween the drawers?

Thanks for sharing it.

THIS youtube discusses one similar.

The market on this furniture is a lot softer than it was a decade or more ago--unless it is a particularly rare piece by a known maker. This falls into the "brown furniture" category and it is possible that this piece could have lower auction value than it would cost to buy a well-crafted reproduction. (Which seems ridiculous). I would certainly not try to sell it at this time, if that's why you are attempting to find current value. It's worth keeping.

Honestly, if these pieces you are posting really do date to when you say, you should be dealing with high-end local antiques experts, not the internet.

There are things you can only determine for sure by actually seeing a piece in person, and the difference in value can be very, very considerable. Don't let convenience put you in the situation of being the one who gave someone the incredible bargain that most people dream about finding.

Does anyone remember the old Bartley Reproduction kits? This is a pic from their 2004 catalogue. 74" high, no carvings, no drawer locks. They said this is a QA Highboy reproduction of a New Hampshire 1750 original. In the old catalogue the price for the finished piece was $4495.

OP what do you think your HB is worth?

HERE is another youtube with highboy appraisal. The beading detail around the drawers is something I always look for. Notice the brasses, they don't look original and for that reason value is knocked down to $8 - 12,000

If you notice right off the bat the piece he is talking about is much older than yours by the texture of the wood in the video sam_md posted. I personally think your piece if fairly recent and unlikely to be very valuable except as a family piece. It is nice to dream but so far what you have posted have been mostly recent copies. With the market as it is for this type of furniture finding a buyer that will pay anything near what it originally cost if probably not realistic. A suggestion that I do have is to find a stager who might be interested in adding them to their inventory for a couple of hundred dollars if lucky.

The piece in the Youtube video has burl veneer....that's why the seemingly uneven look. Not a product of age but of the way it's made.
The OP's piece is solid cherry., he said...but has shown no interior details...however I have seen no repos anything like that....and even if it were I would expect to pay well more than "a couple of hundred dollars".
New furniture...https://www.dutchcrafters.com/Amish-Queen-Anne-Flat-Top-Highboy-Dresser/p/2835?so=1&gclid=EAIaIQobChMI9JaUgJLG3QIVB41pCh1EUQDnEAQYAyABEgIgEfD_BwE
In spite of what you think, people ARE buying "brown stuff".

The market for even authentic Chippendale furniture - i.e. from the factory of Chippendale is down. The market for even authentic real antique furniture is down.

Just because a piece was owned by your grandmother doesn't mean it's old and rare. As others have posted, to establish current market value as well as actual age, it would have to be appraised by an experience person who views in person. As a start you could probably get some kind of initial show of interest by taking very good close up photos of the piece including interiors showing method of construction since stuff like type of screws and nails would be clear markers of age.

That something is antique and was once a certain price has nothing to do with current market value. Cookie jars soared in value after Warhol's collection was sold in auction but prices for even rare ones are now low. Whatever was collected by the parents, grandparents and even the boomers has collapsed in value because it is no longer in style for the most part and it is glutting the market as people have died and no one wants what they owned.

Unless the maker had a unique signature on a piece that is recognized in the trade the piece would need to have been listed in the inventory/will or some type of document for it to be accepted as being as old as the OP states it is given how fresh it looks. I have seen furniture and other things marketed as being "in the style of various dates, styles, etc.". This area had a fine furniture store that specialized in having excellent furniture of many eras. They sold it as being of that era and did not attempt to defraud anyone. Post sale people when reselling claimed that the furniture sold by them was the true article mostly because they did not know the difference. The OP can probably find photos of it being in relatives homes back to the late 1800's but would need some type of proof that it was actually made in the mid-1700's. Doing this can add value to the pieces that are being found.

HERE is another Antiques Roadshow youtube scroll to 9:45 and hear the details of this beautiful highboy. The appraiser makes many good points. Perhaps his best quote: "Demand is the best arbiter of value". After that he gives his appraisal value.

What's interesting is that the top three drawers have key-holes. Do you have the key for these?