Will it affect my tax return if my current mailing address is different from my pay stubs?

The address on your tax return and the address on your W-2 do not have to match.  People move all the time so this is not an issue.

No that will not affect your refund.  The IRS knows that people move and change addresses all the time.   Use your current mailing address on your tax return so the IRS has the right address to use if they need to mail you a letter or a check.