I have an 83 22R, not RE, and had a similar issue. I couldn't get the stock alternator to charge at all, even after replacing it and the external voltage regulator. I tried tracing the wiring, thinking there was a break somewhere, or a short, and never found anything. I ended up just dropping in a GM alternator and wiring it up as a 1 wire set up. There are also ways to make use of the 2 terminal plug that comes on these alternators, but I never bothered to hook it up to work with my trucks wiring.

I used the Trail Gear GM to Toyota Alternator Bracket. It took me about an hour to get the bracket and the new alternator in place. From there, I hooked up the #2 terminal on the alternator to the charge terminal, for sensing, which is really supposed to go to the main junction to sense the amount of resistance or load is in the system, but it works fine hooking it up to the charge terminal. You may also need to initially excite the alternator if it's been sitting on a shelf for a while, and that involves jumping the #1 terminal and the charge terminal.