Being from Texas, what is the time after a divorce that a party has to wait before remarrying?

I stand corrected as Texas does in fact have such a requirement.  In Texas neither party may marry a third party before the 31st day after the date the divorce is decreed. (usually means signed).  However, the former spouses may marry or remarry each other at any time. That is because wither party to the divorce action can appeal within 30 days. There is an exception that when “good cause” is shown the court can waive the prohibition against remarriage if a record of the proceedings is made (in open court with a court reporter) or with “written findings of fact” prepared and signed by the Judge. 

As for what the legal problems that can come from re-marrying within that time frame I can not answer that.  You need to speak with a divorce attorney in your area.  You can check here at attorneypages.com. 

I'm not a Texas lawyer, but my research suggests that you're right about the 30-day waiting period. It applies to any remarriage other than one between the two divorced ex-spouses.

Would you also have to wait that same 30 days before getting married in another state?  I think the answer is probably yes, because typically courts in one state give another state's court's judgments the same effect as the one issuing the judgment.  Here, the Texas divorce judgment doesn't free you to remarry until the 31st day afterward, so most states would, I think, make you wait the same way.

It might be that another state where you went for a marriage license might not ask a question that brought this out.  But if the legality of your re-marriage was questioned later, you could have a problem.