Can I claim my son as a dependent if he is a full-time college student living at home and has a part-time job?

Yes----if he is under the age of 24.   If he is a full-time student under age 24 at the end of 2023 he can still be claimed as a qualifying child dependent.

MY DEPENDENT HAD A JOB
If your dependent has a W-2 for his after-school job, summer job, etc. you do not include the information on your own return. You can still claim your child as a dependent on your own return. He/she can file his own return for a refund of some of his withheld wages (he won’t get back anything for Social Security or Medicare), but MUST indicate on it that he can be claimed as a dependent on someone else’s return. (Supervise this closely or prepare it for him!)
If your dependent’s earnings were over $400 and were reported on a 1099Misc or 1099NEC then he must file a return and pay self-employment tax for Social Security and Medicare.

You might also want to use free software from the IRS Free File versions:
https://apps.irs.gov/app/freeFile/

Yes, if he is under 24 and did not provide more than half of his own support in 2023.

If you need help further help, please comment below and provide more details.