One thing has absolutely nothing to do with the other! First, YES, you should refinance to a 15 year loan. And if your credit is good, you should pay only about 3% interest on a 15 year loan. Do it immediately as rates are rising. Check with your current bank or GuaranteedRate.com or go to Wintrust, assuming you are in the Chicago area.
Second, your bank (or new mortgage lender) should adjust your escrow account to cover the increased property taxes. That would be the same whether you stick with your current loan (though why do that?!) or refinance.
Be sure to pay the escrow shortage immediately!