It sure sounds like your power going in is getting interrupted while driving ,but not while sitting still. An issue like that is almost never in the radio itself .
It sounds like you have already traced your 12v power red wire going in to the radio and inspected it all the way for any signs of corroded terminals, splices or loose connections all the
way up to the fuseblock on the firewall . We do the exact same FM conversions here at Wards Classic Car Radio Repair , probably 500-600 conversions over the years , and
have had one or two customers who had underdash wiring issues that created such a problem as you described.

Greg