If my ex-employer offered me severance pay but later on decided not to pay and sued me stating that they don’t owe me anything, can I provide their offer of severance as evidence that they actually owe me?

It depends on whether the offer constituted a written contract or not. First, it must have been in writing, since employment in this country is employment at will: the employer's right to change the terms or conditions of employment, or to terminate an employee at will without notice or severance, is only modified by a written agreement.
Second, it must, as stated, constitute an actual contact. To be a contract, not only must they be offering you something: you in turn must be offering them something. For example: if there was a written agreement stating that in exchange for severance, you would give up any possible claims you may have against them, or would keep certain information confidential, that would be a contract, you since are each offering the other something of value. If there was such a written contract offering you severance in exchange for you doing something and you accepted that contract (accepted those terms), then you can enforce it against them, via a lawsuit (e.g. suing for the severance) if necessary.
However, if they gave you nothing in writing, or if they gave you something in writing just annotating the severance they at that time proposed to give you but not obligating you to do anything to receive it, there was no enforceable contract and you cannot compel them to give you severance. Enforceable contrats  require that each side give the other something. In this case, they would be free to change their mind and deny you severande if they chose.