It's true for any job switch that you have to find a new employer willing to pay for you.
When I was on H1B and faced that situation, I talked to an immigration attorney who told me how things really work.
Caveat: This was 10+ years ago. I doubt it's changed, but talk to a lawyer if you're concerned. A one hour consultation won't break anyone's finances.
Like I said, talk to a lawyer if you want to know what's really up.
The real problem with the H1B system IMO is the uncertainty and inconsistency of everything which leaves those affected in a constant state of concern.