Hello!

Took the car for a short drive yesterday, and started noticing a thumping sound coming from the rear axel. It only happens when accelerating. And if I accelerate slowly, it doesn't make any noise. Transmission? Thanks in advance, gang.

 

could be many things, a muffler or pipe Hanger, a bad U Joint, a bad Transmission Mount,

if the transmission is shifting ok, then it is not the transmission, most likely, it is the transmission Mounts, they are rubber and deteriorate with age, a 1983 is a candidate for replacement transmission mounts. Get new ones on eBay. They are very similar to engine mounts.

 

It is the transmission mount. This is exactly the symptom of a badly deteriorated mount - when you have it replaced make sure that the mechanic installs it correctly (meaning that he/she has observed how the original was configured).