I've been working at am automobile claims office and apart from
working on property damage and collision claims, I've also been given
Auto Theft claims.

Now I might be offered a full time position as a full-time theft
adjuster, with only theft files, no property damage. I think I have
managed to gain some leverage by paying close attention to my work.

Now my question for you: Do I pursue a raise if I accept the position?
And if yes, how much does Theft Adjuster make in the southern
California area?

Thanks

You should certainly enquire about the salary for your new position
but sometimes it's advantageous to take up a new appointment at the
same salary just to show that you are a good company man.

Only you know the company and only you can decide.

Good Luck whichever route you take.