The above posts refer to the Chrysler ignition/electronics sytem (1991 on). Your 1990 with the Renix system is different, with no power distribution centre under the hood. If the fuses (under the dash) check out OK, AND these electrical items that don't work are on different circuits (different fuses), then your problem is most likely the ignition switch (the switch is the black box at the lower end of the steering column- not the key cylinder at the top). Several circuits feed through this switch, and when it burns out some contacts then circuits that are otherwise seperate quit working- the common factor is the switch. That the electrical items work temporarily while in start key position, but not in run key postion, also suggests the switch. It is a standard GM switch used in a lot of late 80s and early 90s cars, supposedly not too expensive.
If it isn't this switch, it could also be corroded grounds making poor connection, but less likely. You could check the ground strap from the back of the engine to the firewall, and clean the connection points (good idea anyway).