No they can’t use the logo for free. Texas law requires that Texas schools be paid. And the revised Texas NIL law was just signed into law this past summer. It amended the prior NIL law which previously said athletes were banned from using school logos in NIL activities, even if the school wanted to grant permission.
As to Florida, they have the same type of MMR deal as UK, it’s just with Learfield instead of JMI. The Florida NIL Collectives have signed sponsorship agreements with Learfield. Any business who is engaging in a legimate NIL deal with an athlete would also have to sign a licensing deal with Learfield and pay for use of Florida’s logo in any advertisements. And if an athlete wants to promote any co-branded merchandise (e.g., shirts), Florida has set up partnerships with specific vendors that the athlete would use so that the school collects royalties via their agreement with CLC.