Should you apply to other schools (without the intent to play that sport) after signing and athletic letter of intent?
You can apply and attend another school but usually you can’t play that sport for at least a year at the second school if it is also an NCAA D1/D2 school. Some who commits to a D1 school can choose a D3 school and play immediately and I think can also play at a JUCO or NAIA school without penalty.
But really, don’t sign if you aren’t sure. If you need more time tell the coach. I asked D’s coach what would happen if she didn’t sign in the early signing period. The coach was honest and said the offer would still be there but wouldn’t be more money if she waited till spring. She said the biggest benefit was having everything over, and being able to STOP applying to other schools. And she was right. It was very nice to be done!
I don’t know why you’d apply to other schools after signing an NLI. As twoinanddone pointed out, isn’t the point of an NLI to be done? If your kid is unsure about the school or wants to explore other options, s/he should not sign an NLI.