This is a situation that requires real caution, and getting the right information before taking any action is essential.
TDCJ requires visitors to go through an application and approval process that includes identity verification. A foreign passport is a legitimate form of identification and is generally accepted for visitor applications at Texas facilities. On that level, having a passport from your home country rather than a US ID is not automatically disqualifying.
The concern in your specific situation is what happens during that application process and what information it surfaces. Visitor applications involve background checks, and the current environment around immigration enforcement means that any interaction with a government-connected system carries risk for someone without legal status. Correctional facilities have relationships with law enforcement at multiple levels, and a visitor application is not a protected or confidential process.
Before doing anything, call Pam Lychner's visiting room personnel directly and ask about the process without providing identifying information. Ask generally what forms of ID are accepted and what the application involves. That call gives you useful information without creating any official record.
You should also speak with an immigration attorney before taking any steps toward a formal visitor application. An attorney can assess your specific situation and advise you on the actual risk level given your circumstances. That conversation is confidential and protected, which makes it the safest place to get real guidance.
The desire to see your husband is completely understandable. Just make sure the steps you take to get there do not create a situation that makes things significantly worse for both of you.