Comparing the datasheets for the 5ADP- and 5CP- shows MULTIPLE differences in required element voltages. The 5CP- uses a post-accelerating electrode, while the 5ADP- doesn't. They ARE NOT the same tube, or even close substitutes for each other. I can't see how a single scope could ever accommodate both tube types. The differences are just too great. Are you just leaving the accelerating anode lead hanging loose when installing the 5ADP-?
According to the manual I have here, the 304H is designed for a 5CP- type tube. The "H" designates High Brightness, thanks to the post accelerator. You need a 5CP- tube. Fortunately, they are VERY common, and relatively cheap. Here's one for $25 Buy-It-Now:
http://www.ebay.com/itm/Vintage-RCA-...item1c37c1489e