Comments based on 25 years of using and selling AMSOIL synthetics:

Yes, change to synthetics. I recommend using a good engine flush first (AMSOIL and BG come to mind). If you don't, or won't, flush, plan to change the oil filter after 2,000 miles, and the first batch of synthetic after 4,000 or so.

A good PAO based synthetic will easily last 10,000 miles in a sound engine, but I'd change the filter after 5,000 (or 6 months).
A non-PAO "synthetic" should be good for about 5,000 miles, maybe more if you take oil samples and find out for yourself.

You don't have to change the transmission, but it is a lot less expensive than buying a new one. Synthetic ATF will drop the operating temperature anywhere from 20-40 degrees F, and last at least 3 times longer than non-synthetic. Do a search and find the recommended do-it-yourself instructions.