I just had this EXACT SAME PROBLEM and I fixed it. Went to get the van inspected for tag renewal and Junior says that one of the tag lights doesn't work. Says he can fail it and I can fix and come back for additional charge or Junior has the bulb and he can fix. Chose door number 2 so I didn't have to come back.
Driving to SC early in the morning the next week and notice I don't have any instrument lights on the dash, ok who was the last idiot to work on van, Junior, and he replaced the license plate bulb. Experimenting, I find that if I turn key on without starting, lights work, but as soon as I start it, lights go out. Check license plate light and Junior has broken the bulb holder and tried to jam it in there. It is loose and could easily short out. License plate light is on the same circuit as the lights.
It did short out and the fuse is under the drivers seat. It is fuse number 8. I replaced mine and it shorted again and blew the fuse. Junior broke the license bulb bracket so bad it is shorting. Replaced the bracket and fuse and all is well.
Also, I could hear a relay clicking under the dash which is on the steering column. Check that relay. When I started the van, I could hear that relay activating.