You are right in that you have no way of knowing if the school that hires you is going to be bribing officials or forging a degree on your behalf, and, to be honest, schools don't do those sorts of things unless they know they're going to get away with it.
Of course you need to be up front about the fact that you don't have a degree so that the school can give you a straight answer.
I will say that if you already have a proper residence permit, it IS much easier just to have it transferred over to the new employer (with a release) than it is to do the whole thing again from scratch. This may be what some of the schools are alluding to. I am not absolutely sure, but I think once you have the work visa/RP already, when the new school applies for a switch they do not have to show all of the original documents all over again, at least if the switch is within the same province. Again, I can't say this with 100% certainty, but when your contact said it was easier to extend/renew, that might be what she meant.
To be honest, I don't think there are any provinces at all that don't officially require a degree to work legally. Whether or not those rules are enforced or not is an entirely different story, but the trend over the years has definitely been towards a stricter interpretation of the laws.
And look, I'm not going to advocate working illegally, but you're married so you need to stay in the country, going home is not an option. If worse comes to worst, get the spousal RP. The process is pretty simple and at the very least it will buy you some time. I am sure you'll be able figure something out.