What are your thoughts on the matter?
My thoughts? Pretty simple - I can tell you that on both the Croatian (EU border) and Romanian borders, they're scrutinising passports very carefully at the minute. You're not going to be able to get away with a visa run there - the Hungarians are treating the Romanian border as if it's a non-EU border for passport control purposes. Likewise, the countries surrounding Croatia are being very strict at the minute - I was stuck in a 2 hour queue just to leave Slovenia a few weeks ago.
By making such a run, you've got no guarantee of success - in fact, the chance of failure is quite high. Despite what's said online, the national border police are making quite a concentrated effort to deny re-entry to obvious border-runners- and do you really want to be stuck on the Romanian/Croatian border with a refusal from the Schengen space?
Then I decided to move to Warsaw on a promise from a school to teach Business English for a particular wage and specific number of hours. Well, I took the promise at face value
First rule of business : do nothing without a contract. Applies everywhere in the world.
Sorry to say, but as a non-EU citizen, you have certain responsibilities - and no-one to blame but yourself for not applying for a residence permit. As Harry says - the best bet is to get out of the EU alltogether, stay out for 6 months, then get a new passport and come back in. Even then - there may be some questions asked about what you've been doing between your stays in Poland.