Odds are they are both just dirty. I know, you cleaned them, but paint gets in places and dries before you know it. Personally I'd disassemble them both and soak all the metal parts (all of the Badger 200 EXCEPT the air valve, I don't know about the Paasche) in lacquer thinner overnight. You'll be surprised how much paint you didn't get out.
As to the Badger air valve, odds are soaking the body in lacquer thinner will clear it. There is an air passage bored through the body that allows air to flow from the valve to the tip. Sometimes paint will manage to get in there and plug it. If not, remove the air valve from the body (it unscrews), use a hex wrench of the appropriate size to unscrew the bottom of the valve (where the hose screws on). Carefully remove the cap, spring, and plunger from the valve body. There is a captive o-ring on the plunger. Clean everything EXCEPT that o-ring with lacquer thinner. For the o-ring and plunger just use some alcohol on a Q-Tip. I usually put a tiny bit of airbrush lube or Vaseline on the o-ring, reassemble the valve, and it will be good to go for a while.
If that doesn't fix the problem with both of them you've most likely got an air supply problem.