Honshu west coast by train

Is it possible to travel by train (preferably slow trains) from Aomori, south along the west coast of Honshu?
Passing through Akita, Sakata, Niigata. And if so, would one purchase a JR pass or would that be false economy for this specific train journey?

Yes, it is possible and a wonderful thing to do if you like slow train travel. There are limited express trains that are relatively fast. And there are local trains which are very slow and have very bad connections along the way. Aomori to Niigata by local trains takes about 10 hours. But if you like that type of travel, it can be highly rewarding.

South of Niigata (between Naoetsu and Kanazawa), the coastal train line is non-JR and not covered by the Japan Rail Pass.

The Japan Rail Pass will not pay off for that journey alone, especially if you use local trains. But if combined with other trips, it may pay off.

The Seishun 18 Kippu is good ticket for that type of local train travel:
https://www.japan-guide.com/e/e2362.html

It's possible to travel almost all the way along the west coast. As mentioned by Uji, it's reasonably slow but very scenic. For long distances, it's only single tracks and quite windy (not straight) which also means that there are not too many trains.

Uji is also right about train tickets are a lot cheaper on the west coast. Also note that the west coast has much fewer tourists but still heaps to do and is much more traditional and slower as a result. This is specifically true of the west coast of Chugoku.