Do you have the sentry (gray) key system? If so, is there a red dot flashing in the cluster?
I've seen the sentry key module fail in these vans before causing these exact issues. Factory alarm problems can also do it. In the case of the former, the fix is to either get the dealer to replace the module and reprogram, or re-solder the board in the module. Note that the module makes extensive use of surface mount parts... best to let a professional do the work.
When my father's old 2002 Caravan did this, it could sometimes be made to work again by wiggling the wires under the dash. I did the re-soldering myself using my hot air rework station, and the van never had another problem.
I've seen the sentry key module fail in these vans before causing these exact issues. Factory alarm problems can also do it. In the case of the former, the fix is to either get the dealer to replace the module and reprogram, or re-solder the board in the module. Note that the module makes extensive use of surface mount parts... best to let a professional do the work.
When my father's old 2002 Caravan did this, it could sometimes be made to work again by wiggling the wires under the dash. I did the re-soldering myself using my hot air rework station, and the van never had another problem.