That model you have likely has the A4BF1 or -2 type trans, if you changed a filter in the bottom pan that would confirm this, as the other trans has the pan on the side of the unit and the filter isn't acessable without splitting the trans (with the exception of a few v-6 models that have a 2nd "spin on" fliter additionally). The A4BF1 has a pretty common problem of the direct clutch drum laying down and galling up on the pump stator, destroying the teflon sealing rings between them that transmit the oil pressure to the direct clutch which is used in reverse, 3rd, and 4th. It is a common enough problem that the parts are very difficult to find and expensive. Later revisions have changed the bushing layout in the drum and are more reliable.
As far as solenoids,sensors, etc I don't think it is likely as reverse has no solenoids in the circuit and runs straight off the manual valve(shift lever), in most modern transmissions except for some VW and Ford CD4E reverse will work fine even with the connectors unplugged. Forward will engage but usually be in 2nd or 3rd "limp mode" The computer actually puts the trans in limp by just turning off all the solenoids and 3rd is usually where the shift valves "rest" when solenoids are off. It probably isn't the torque converter...think about it...the engine doesn't spin backwards in reverse so the convertor just inputs power in clockwise the same as it does in every other gear so the issue would affect all ranges if it were the convertor. The convertor has no idea what range the trans hooked to it is in.
You are probably best off looking for a used unit if you are on a budget, as the pump stator and drum are usually difficult to find and expensive as these are the most common failing parts(the trainsmission industry usually uses inspected used or rebuilt parts as many auto manufacturers refuse to sell individual parts and want you to fork out 3-5k for thier reman unit(don't get me started on how i feel on manufacturers not making parts available!![:@]lol)) The used parts deal is not a quality issue usually on hard parts like drums as its like a crankshaft or something that doesn't really "wear" it just either blows up or is just fine with new bearings.
How do I know this? I've been one of these trans shop "Sharks" for 24 years.
Don't be too hard on them, as long as they aren't selling you something you don't need or gouging you, offering to tow your car and get it there to check out isn't any different than anyone else trying to get customers to service, please and make a living. Of course there are bad apples in every batch I won't argue that.
Not much testing you could do at home for this unfortunately, but I will say the fact that there arent any solenoids required for reverse operation strongly suggest a internal mechanical failure in either the low/reverse or direct clutch which would require unit disassembly. The delay into gear and lack of movement otherwise is probably from the debris blocking the filter after a few minutes and not allowing the pump to get fluid to even the forward clutch. I'm betting after you shut it off it would move forward for a little while until the filter gets clogged with debris rising up from the pan? The not upshifting is because when the unit tries to shift to high it needs to use the same clutch that is used in reverse and that clutch cannot "grab" until you let off the gas and let it slowly couple the drums together with what little force it has left. in reverse the trans is actually bound in 1st and 3rd at the same time and requires a lot more "grab" than does it in 3rd, which is why you get 3rd "sometimes" , but reverse never
Sorry I don't have better news, but best of luck...Todd