I doubt that OP is still active

You might be able to get more help for your specific car in the RX model forum for 99-03:
https://www.clublexus.com/forums/rx-...1999-2003-182/

If you're having something similar, I would suggest to very carefully check your wiring for issues. It may not be as simple as plug & play with a fuse

 

My Lexus ls400 got same issue, intermittent car starting without dashboard lights and can’t shift out of park. Had it looked by the auto mechanic and sure enough it was the alt fuse 120amp.
I was worried my alternator was getting short or fried from the slow power steering pump leak, but luckily the mechanic put a a curvy flap to direct the leak away from the alternator.

Above the car battery closer to the driver is this fuse box

Replaced this 120A ALT fuse.