Yes, you can definitely get married in Texas despite the fact that he is out of status (illegal). In Texas, you would get married like anybody else would. You would go to your nearest county clerk's office together and you have to show your identity to prove who you are and your age. For his identify he can show his valid Guatemalan passport.
Unfortunately it sounds like he might have an order of deportation if he was supposed to show up to court and never did. He should call the court where he was supposed to go to get this information.
If he does have this order then once you are married, you should submit the I-130 petition for him which you can find on uscis.gov/forms Read all the instructions on what you need to include with it. Once that is approved, he would need to reopen his immigration court case. It is very recommended that he get an attorney for that because it is a delicate and complicated process. Once the case is reopened, the attorney will guide you on how to close it correctly and then he will have to apply for an I-601A waiver since he entered the US illegally (without a visa).
You can read about the I-601A waiver here https://www.uscis.gov/family/family-us-citizens/provisional-waiver/provisional-unlawful-presence-waivers
I know immigration issues sometimes seem complicated and confusing. If at any time you would like to have a more detailed consultation by phone or if you need help with application preparation or review, I can send you an offer for this legal service to you for a reasonable fee later on. You don't have to decide if you would like that now, but just letting you know it's an affordable option for you to keep in mind.
Good luck to you during this process and I hope this answer has helped you!
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