I’m trying to see how selling a paid-for house will improve your situation. You say you can’t afford the taxes on a house in the “lower class” section of a wealthy town. Assuming you sell and have enough for a down payment on another house, won’t you be paying taxes on a better house, in a better part of town? In addition to a mortgage? How is this going to work?
If the taxes are the main issue, I’d take a serious look at appealing the valuation.
You work with a specific realtor, not an entire office. If anyone you interview acts as if she is laughing you off, go elsewhere. I’m not a realtor, but I have sold a house in one day. My advice about where to start in picking one is to drive around your neighborhood and look to see who has most of the listings. Also, who do you know through marketing to you? If they succeeded in making you aware of them, they might do the same for a buyer. We like the agents we used for buying our house, and stay in touch, but they specialize in another part of town and so I don’t think we’d use them when we sell. (They’ll probably be long retired by then anyway.)
AuntiePam said it doesn’t cost you anything to put your house on the market. I’m not sure I agree. First, there is the cost of keeping it fixed up so that you can scram whenever it is shown. Second, you don’t want it on the market until it is ready to show. We got a very good deal on the house we bought because the owners put it on the market, decided to do some fixup work, and more or less kept it off the market until they were done. The days on market was very high. On the other hand, when we were selling in NJ we painted, and got a storage area to put all the clutter in. The house had just enough stuff to look lived in, looked fresh, and sold the day of the open house. So, throw your stuff out (or at least hide it) first.
In this market I agree you should have a buyer lined up before you buy another. And, when you buy, make sure you have a contingency clause letting you out of the contract if your sale falls through, if at all possible. And don’t let your buyer have one if you can.
One other thing that I don’t remember seeing: if you can’t afford upgrades, make sure the place is spotless, empty of clutter and “collectibles”, and at least has a fresh coat of paint inside (and possibly out). A realtor can help you with paint colors that help sell. Cheap carpet isn’t a huge outlay, either.
Yeah, a house that’s sitting on the market is getting stale - people are interested in the new listings, and expecting you to drop your price for a stale listing. Our previous house sat on the market for months and months, and the price kept dropping - we offered even less than the last price drop, because it was obviously not a house in demand. We sold the same house at our first open house because we had it ready to go.
As to pricing it rock-bottom and getting it sold, I guess that would probably work, but it offends my frugal nature. 
I hate not maximizing my profits, if I can get more with some elbow grease, but it depends on what your goals are - just sell the place, or make as much money as you can selling the house (without getting into serious renos).
One danger to a rock-bottom price is that people will expect something to be seriously wrong with it and might not even come look at it. Then they will probably low-ball you even with a rock-bottom price - you might want to leave yourself some negotiating room.
Oh, another thought came to me - having a lower-end house in a higher-end town is a very good thing - people want to live there, but not everyone can afford it. Your prospective buyers might be people who want the address but aren’t loaded to the gills.