Configure your multi-meter for current(amps) tests, and connect it in place of fuses in the fuse-box to find out which circuit the problem is in. Once you know which circuit he problem is in you can start debugging it.
Likely something in the light/Dome fuse circuit but just a guess.
Did you have this problem before the alt/battery replacement?
To protect your battery do the test as soon as possible, and once you ID the circuit leave the problem fuse out unless you are driving the vehicle.