I think it's good you have the vet appt.
Let us know what the vet has to say about all this.

As for transporting when I take my aquatics a short distance, I consider a 45 minute trip to be a short distance...
I put them in a plastic box on a damp towel then cover it all. When you cover them up so it's dark in the container they are more likely to calm.
Good luck!
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PHRatz