Ok. Well, I can address your question just with the information concerning the lack of a date.
Here's the problem. There used to a claim called "promissory estoppel" which essentially means reasonable reliance. It could be used to sue an employer for not giving something that they had promised and the employee had reasonable relied on, to their detriment.
The courts took that claim away though in your state. It is no longer recognized in "at will
" employment situations.
So, everything swings here on what type of contract you have. It doesn't have to say anything specific to make you "at will." That is the default position, so it has to say something specific to make you something other than "at will." If your contract doesn't specifically state that you can only be terminated for cause, it is an "at will" contract.
If it is an "at will" contract, you have no claim against the employer because "at will" employment can not be the basis for a "promissory estoppel" claim in your state.
If it is not an "at will" contract, then you're still not in the greatest position because the contract doesn't state a start date. Certainly, after a "reasonable" period of time (and yes, that is a very slippery term), you could sue them for not honoring the contract, but that is really going to depend on how long you've had to wait. Two months is a long wait, but at least one month of that is going to be considered "reasonable" if not 6 weeks. You'd then be suing over two weeks of lost wages
and that suit would also result in your losing that job, because it is not a protected type of lawsuit (like suing for race discrimination