I would check on the possibility of placing a public notice ad on the local paper informing her of your moving plans. I think after three weeks of running the ad you can officially claim that you did your best to notify her and you will have proof.
If she owes child support she probably won’t even want to respond for fear that she will be charged and forced to pay. There is usually a legal procedure to be followed. You need to modify visitation – file the appropriate papers for modification and then file a motion for notice by publication. The court will tell you how many weeks (usually 3) and the papers to put the notice in.
Also, if you really want to cover yourself, give the papers to the local sheriff (or whoever you folks use to serve legal documents) and the mother’s last known address. Get an affidavit from the sheriff documenting his efforts to serve the papers.
And that, as they say, should be that.