Have a burgundy 1988 300zx Turbo GLL 2+2.
Great car, very reliable. Bf and I went for a ride once and all of a sudden it started to overheat, didnt know what was wrong til we came home and realised we blew a belt... but even tho it run VERY hot it was still driving awesome.
Took it on a roadtip thru Cali and Nevada too... handles the heat very well. Got great mileage even with AC running or windows down.
Its a pig on gas, yeah but only in the city... lol
They are build for road trips... highway driving and stuff like that.
Hard to kill... unless something is very wrong with the harness (yes, I have experience with that)
Your tranny problem: Did you check the transmission fluid level? Make sure the engine is idling and all the way up to temperature... might be the clutch bands slipping if its not the fluid level.