I'm not a fan of E-H branded Russian 12AX7s. Or any other Russian 12AX7s for that matter. But tube quality doesn't have a lot to do with whether they will stay clean. A modern amp with cascading gain stages in the preamp is designed to allow you to dial in preamp distortion. A quality preamp tube will certainly provide distortion under those circumstances. If it didn't, it would be putting out less than spec output, and so it wouldn't be a quality tube.
All you need to do is not dial in any distortion in the preamp - how exactly to avoid that depends on the design, but basically, given the choice, you want to select the channel with the least number of preamp gain stages (not counting tubes or individual triodes used for post EQ makeup gain). Then, if you have separate controls for the later gain stages, you want those off or as low as possible, and you want to use the first one as your real volume control. The master volume pot - which is, in nearly every case, in reality part of the preamp circuit and not the power amp - should be all the way up - unless the amp gets noisy when used this way, as some designs do, in which case you'll need to split the difference and back it off.
High input signals (whether due to active pickups, onboard preamps, hot passive pickups, or pedals) will make this harder.
If you're doing all of this and it's still an issue, you might try subbing some of the preamp tubes for lower output equivalents. There are lots of threads that go into this. Personally I favor clear top RCA 12AU7 for this purpose, which do the job, are very common and last a very long time. Sometimes a 12AT7 will work better. Impedance as well as gain differences come into play with these different dual triode tubes that share the 12AX7 pinout.