What to do if my husband and I went to the court and signed common law marriage papers but now that we are estranged he won’t help pay for the divorce because he thinks the marriage no longer exists?

You have a couple of different issues.  The first is whether or not you have a valid common law marriage.  To know whether or not you had a valid common law marriage, take a copy of your informal marriage license to an attorney to be reviewed and go over the facts of your case.  If you only filled out the license, but did not conform to the requisites of an informal marriage, then you may not need a divorce. 
Your second issue is whether or not you actually need a divorce.  Not all informal marriages require a divorce.  (This is not the case with formal marriages)  The Texas Family Code has a provision which enables certain common law divorce actions to avoid divorces.  Your's may be one of them.... so before you spend the funds on a divorce action, see if you actually need one.
Your third issue is regarding what the clerk will tell him.  The clerk's don't give legal advice and they don't cross reference records before issuing new licenses.... so they, most likely, won't tell him anything unless he asks them.
Your fourth issue is the cost of the divorce action.  The court's don't automatically split the costs of a divorce action.  This means that you could potentially be required to absorb all the costs of the divorce action, even if he doesn't contest it.
If you don't have any major property items and you didn't have any children together--then the divorce action should be simple, assuming that you need it.  Many counties are not offering clinics where you can have an attorney review your case and documents to provide you with the answers that you need to your questions. This will significantly reduce any costs.  Call your local district clerk for information about clinics or pro bono programs in your area.