Texas has no income tax on individuals or corporations. However, other states that have an income tax can tax you on income you earn from work you perform within their borders. You will have to file a non-resident tax return in any state in which your income exceeds that state's filing threshold. You can find a state's filing threshold on its state tax website.
Are you the only member of your LLC, or are there other members? A single-member LLC is disregarded (ignored) for income tax purposes. But if there is more than one member the LLC has to file a partnership tax return. If the company that you work for has a contract with the LLC or pays the LLC rather than paying you personally, you might have to file partnership tax returns in the states where you worked. If you have a multi-member LLC you should consult a local tax professional who specializes in multi-state taxation.
An LLC, single-member or multi-member, can file an election with the IRS to be treated as an S corp or a C corp for tax purposes. Has your LLC made such an election?
Does the company that you work for pay you as an employee (you get a W-2) or as an independent contractor (you get a 1099-NEC or 1099-MISC or no tax form)? Some of the states that you work in might have different rules for an independent contractor or an LLC than for a W-2 employee. If you are an independent contractor then, again, you should consult a local tax professional who specializes in multi-state taxation.
Here's the TT link to State Filing Requirements:
https://ttlc.intuit.com/questions/2903200-do-i-need-to-file-a-state-return