Yes, Rice is blind but not all the other are nor do they all meet full need.
If your need isn’t outrageous, you’ll be fine. But if it is, you never know. An article was published about meets need Lafayette. They were open to the reporter and didn’t admit many kids they normally would have. The budget is only so much. And they have to spread it.
I don’t know your level of need.
I have a good feeling for you at Rice but everyone needs an assured AND affordable safety. Rice is not that. I know you have admission safeties!!
Do you have that ? Is UT affordable - room and board etc?
Best of luck.