OK so we know it's a chain/sprocket issue. Based on the description and the age of the chain, I'd guess the chain is stretched (worn out). You can confirm this with a 12" ruler. Put some tension in the lower loop by pulling the RDs lower pulley back slightly. Measure 12" (24 links) of chain. Since the chain is 1/2" pitch all pins should line up at 1/2" marks, and 12" should measure exactly 12" pin to pin on centers.

However wear at the pins allow the links to pull apart a bit, and with the accumulated error will have the last pin visibly beyond the 12" mark by a bit. If it's near 1/16" off, replace the chain before it accelerates wear on your new cassette.