To answer your question: find another lab, ASAP
I have a question (which is not related to your post): what can one do/think/expect from a PI who hired you without an official offer letter yet says he's funding is not clear at the moment? (when that "hired" postdoc already turned down a several positions with already funded projects)
Would be great to hear your and commenters opinions.
First of all, being "hired" without an official offer letter is NOT being hired.
That is what we call a verbal offer, which is NOT the same.
Second, do not EVER make plans based on a verbal offer. INSIST on a written contract before you make any plans.
Third, if you already made this mistake, and the PI is disorganized or disingenuous and jerking you around (good intentions or otherwise), get out NOW before you waste any more time.
If, in the worst case scenario, you already moved to a new university, try to find someone else in town if you don't want to relocate again.
If not, call everyone else back and tell them quite honestly why you picked that lab (seemed like a good fit for what you wanted) and that you really had a hard time choosing, but that the funding fell through on the one you picked and you're really hoping the position(s) you were offered before are still open.
In other words, suck it up, apologize, and beg if you have to. This is assuming, of course, that the other offers you had are actually good ones for good labs where you'd be happy to work. And you better hope it's not too late or that some of them have additional funding/enough interest in you to shuffle some things around and take you anyway.
My guess is that you thought PI #1 was a nice guy, and that's why you picked him, whereas the other labs might have been more organized and productive and potentially better for your career, but you didn't like the PIs or the location as much.
Take this as a good chance to avoid making a major mistake. "nice" is actually NOT the major defining quality of a good MENTOR or BOSS. The #1 thing you need in a PI is someone who is responsible, and #2 is someone who is going to keep your career prospects in mind and always communicate clearly with you. This PI has failed already on both counts, so I say GET OUT.
If a PI screws up on something as big as hiring a postdoc and making sure to have money to pay them, you can bet they're also going to screw up on even bigger things, like staying employed and funded in general, not to mention making sure YOU stay employed and have ample opportunities to succeed.