Here are my EXACT recommendations for your situation.
The FIRST thing you should do, based on your wants out of that car, is to ditch the stock tranny and motor. I will assist you in parting them out and selling them. The motor has apparently got some bad internals, but will still bring some good $$$ because of the rareness of a Buick 350. Your Turbo-Hydramatic 350 3-speed transmission will be worth a lot, because that is an extremely popular hot rodders' transmission, because of it's stoutness and simplicity.
Unless you want to make this a project only type car, I would ditch the transmission. The advantage of the TH-350 is that it will take most anything that you throw at it, big block V8's included. So, you could build something like a mean 383 or a small block 400 or 406, or if you wanted to get really crazy, a big block 427, 454, 502, or 505, even a 572, and the transmission is going to hold up with little or no modification. And if you decide to build any of the motors I've just suggested (except maybe a 383), you've thrown your gas mileage out the window in favor of insane amounts of torque and horsepower, so the main disadvantage of this transmission is no longer a disadvantage. If you would take this route (project), I'd say to buy a cheap daily driver, and keep the beast as a "weekend warrior at the drag strip/street killer on nice days" type of car, where gas mileage isn't going to matter much cause it's not a daily driver.
NOW....
If you want that to be a feasible daily driver with some performance, I would sell both tranny and motor, and invest in a small block chevy and a Turbo-Hydramatic 700-R4, 4 speed overdrive. To be exact, I would get a L98 (350 Tuneport Injected V8 used in IROC-Z28 Camaros and 80s Corvettes). It's fairly cheap and makes huge amounts of low end torque, while still having all of the advantages of port fuel injection. Those advantages being that it's friendly on the MPG and will run the exact same no matter how bad the weather is, unlike a carb. Then I would get a 700-R4, which will be the single biggest factor on your fuel economy. If you do a good job looking on eBay and the
www.thirdgen.org classified ads, you can likely score both for $500 to $700. We would probably have to drive to wherever you bought it and pick it up. We could take the Cutlass, as I can fit a transmission in it's trunk, and you could rent a small open-bed trailer from UHaul to strap the motor to, and I could tow it with the Cutlass. I would allow like a good 1 to 2 weeks at minumum for installation with you and me doing the work. Other expenses involved in this swap are a new high-pressure electric fuel pump, a cherry picker rental for lifting the motor out, a new length driveshaft, possibly motor mounts from any donor vehicle with a small block Chevy V8, and a polyurethane transmission mount for a 700-R4($10-$15). At worst case, the bill for the rest of these is going to be like $500, with like 50% or more coming from the price of a fuel pump. So you're looking at a $1000- $1200 budget for the swap. If you take this route, this would be a summer project we could do. I'm currently hacking my way through essentially the same type of swap, as I'm cramming a small block chevy 350 and 700-R4 in the the Buick/Olds based Cutlass of mine, so I have some good insight on exactly what we're gonna need.
Selling the entire car straight out is not going to get you the type of money you'd get parting it out, and as far as fixing the car up, it has the unique coolness of being from a lost era of automobiles, one that was better in many ways to the current one. It's from the muscle car era, and it's a car that would get you onto the Hot Rod Power Tour without a second question, if you fixed it up. Not suggesting that you neccesarily want anything to do with the Hot Rod Power Tour, but if you painted it and put a nice motor in it, it's gonna be respected as a classic.
However, if you really want a classic that bad, there are a ton of other platforms that would be more advantageous. You're not going to find another car from that era that's in as good as shape as yours that's anywhere close to as cheap as you got that car, but there are many other platforms that you could build on instead.
So basically, it's a matter of whether you want a classic, and if so, how loyal you are to the car and to saving money, which considering you paid $500 for it, you did.
If you want to keep it, I'd get a daily driver and make that a project if I were you. The L98/700-R4 swap wouldn't be that bad, but the car is only going to get more and more wear from the elements of daily driving, and that's not a good way to preserve it, obviously. Basically, that motor and tranny wouldn't give you reliability problems, but you're going to wear out other things on the car, and by exposing it to the elements (as opposed to garaging it or keeping it under a tarp), you're only going to give the body more of a chance to rust, unless you fork out $$$ to get it painted. You could always still do the L98/700-R4 swap, and keep it as a nice weather or long trip cruiser, like what I'm doing with the Camaro. That way you still get good miles and enjoyment out of the car, and still manage to keep it sheltered and better preserved.
If you decide to not keep it, I'd part it out, I'd dissassemble EVERYTHING (drivetrain, powertrain, interior, body, everything) and part it out, as your gonna make way way way more money that way, and I would look at either a early-mid 90's GM V6 2 door (like a Grand Prix), or a 3rd generation (82-92) Camaro/Firebird. Both are extremely plentiful and cheap these days, and are good reliable platforms that both will get good MPG while making good power. The F-bodies (Camaro/Firebird) are going to be more on the insurance side, but then again you can do all sorts of things with them performance-wise that you can't with the other platform I listed, and they are much better looking and more respected cars. I'd get a free qoute from
www.geico.com on any car you consider, they are flat out the cheapest on insurance and will give you more of an idea of exactly what you can afF**d.
Honestly, you could probably just get rid of that tranny and motor and probably get at least a grand out of them parting them out, and keep the car as a "down-the-road" project car, and still be able to afF**d the cars I just listed. Obviously, you wouldn't insure it or liscense it until your done, but it would be fairly easy to move, as shedding that particular motor and tranny is gonna drop you around 700 pounds on the curb wieght.
That's my advice!