Can I be jailed in Texas for not paying my debts?

Not legally, but it has been known to happen. Some circumstances where it could happen:

(1) If you are sued for a debt, and you ignore the lawsuit completely. A judgment is entered against you, and the opposing lawyer sends you "discovery," questions about what you own, and requests for you to produce documents like tax returns which would show the lawyer where your income comes from. If you ignore all of this, and eventually the lawyer has a judge order you to answer, but you still refuse, you can be held in contempt of court and jailed, not for owing money but for not obeying the court order and responding to the discovery questions. You can be jailed until you answer the questions, or up to six months as punishment for not obeying the court order.

(2) Also, if you fail to pay certain fines such as traffic fines, a judge may order you to jail in lieu of paying the fines. This is not legal in Texas, if you are "indigent," i.e. broke and you can prove it, and judge must give you the opportunity to do community service instead of paying the fine or going to jail.