I have no idea how to get to what is in your sample, but what I get out of various TB postings is this (and fellow TBers, PLEASE correct this if it is wrong, because I am certainly an entry-level student here):
In your case, if you want less overdrive or more controlled overdrive, you need to reduce the gain in the pre stages, as per Brent Norton and others.
To do that, the 12axt's (mu~100) should be replaced with lower gain tubes such as the 12ay7 (mu~45) or 12at7 (mu~60). Possibly even the 12avt (mu~41-47) although I haven't seen that particular one mentioned, so I don't know if it works well in preamps, as the 12au7 (mu~20) doesn't. So specifically, putting in an EH 12ax7 will probably NOT do what you want to the overdrive.
Another benefit of replacing the existing tubes is a change in the sonic performance. Many tube vendors, like Eurotube and thetubestore.com, give reviews, opinions or descriptions like "super low noise", "rounded bottom with smooth mids", etc. so you can choose a brand & vintage that helps get the sound you want.
And for whatever it's worth, that is about the extent of my knowledge. I sense a frustration on your part, but I think you are getting close, so hang in there.