Note that you remain in F1 status for 60 days after successfully completing your program; this is the "grace period". So you are in status and perfectly fine even if you happen to run into a CBP checkpoint.
However, you will not normally encounter anyone with authority to check your immigration status during domestic travel in the US. The TSA does not check immigration status, nor do TSA officers have the training to understand immigration law; they only check your ID (e.g. a passport) to verify you are who you say you are, and check your belongings for security threats.
There is a small chance of encountering CBP interior checkpoints if you travel in roads or airports within 100 miles of the border, but as I said above, you are in status and have no problems even if you go through such a checkpoint.
(Note: the validity of the visa is irrelevant as US visas are only for entry.)