Here's a good rule of thumb to tell if there's a problem with what your employer is saying.
If you are not getting any calls for interviews, it is virtually certain that your former employer has nothing to do with it. Despite what many people appear to believe, NO employer contacts previous employers prior to deciding who to interview. No one has the time or the resources to do that. Nor is there any reason to believe that your employer is somehow finding out where you are applying and calling to sabotage your employment, as other people evidently appear to believe - first, in 25 years I have NEVER had an employer call me to do that and neither has any other HR person I know, and second, if ever one did call to tell me you would be applying to me and I shouldn't hire you, you'd be the FIRST one I'd call to set up an interview with, if only to find out what the heck was behind that most unusual phone call.
If you are being called for interviews, the interviews go well, they tell you that you are a prime candidate and they'll be checking your references, and then you don't get an offer AND this happens multiple times, then MAYBE you have a problem with your references. Not definitely, because few employers only check references on only one candidate. Personally I check them on my top three, so two of the three aren't going to get the job anyway. (Some do only check one, but it's less common.) So the odds are that you won't get probably at least two out of three jobs even when you do get your references checked. But MAYBE in this case there's a problem.
Patty is right that six weeks is not nearly enough, in the current market, to assume there's a problem in any case.