Best thing is to check out the website below and see if your symptoms are the same... follow the instructions (email him with a list of the symptoms and what is NOT working) and he will get back with you on next steps if it is a GMIII (GMIV or GMV) issue.
Based on what you've stated already, it most likely is. Cost is relatively inexpensive for the repair, and as you will see on his website, most of the time the issue has nothing to do with the key... so tearing into the key and possibly hosing something up may only create more problems.
https://www.bmwgm5.com/bmwgm5/
Hope it helps.