If I just relocated for work and got a cost of living salary increase, do I have to pay more child support even though the increase is due solely due to my new cost of living?

Yes.  There is no exception here.  If the other parent requests a modification of child support-- then you will pay according to your new income, including the cost of living increase.  Income is income--whether it relates to a move up the ladder or a cost of living increase.

Best of luck.