Yes, your fiance is right. In virtually every state prison system, the visitor is responsible for initiating the approval process, not the inmate.
Before you can visit anyone at the Huntsville Unit or any Texas Department of Criminal Justice facility, you have to submit a visitor application. Once that application is in, your information gets run through the National Crime Information Center database, which is a federal criminal records system. That check is standard and it is thorough.
If your record comes back clean, approval is typically straightforward. The process takes time so do not wait to get started. Submit the application as soon as possible so you are not sitting on the outside waiting while your fiance is waiting on the inside.
The part worth knowing upfront is that a felony conviction on your record will almost certainly result in a denial under the standard approval process. That is not the end of the road, but it does require an extra step. The only path to approval with a felony on your record runs through the warden's office. A special waiver can be requested, and while it is not guaranteed, it is the one official mechanism that exists for this situation. Go directly to the warden's office and ask about the waiver process if that applies to you.
Get the application moving now. Visitation approval can take several weeks, and the sooner it is submitted the sooner you can get in front of him.