I would definitely check the rest of that signal path. I’ve run Stax and vinyl and it can be simply sublime, but Stax amps push out a ton of gain (almost as much as an MC phono stage, 54 - 60 dB typical), at very high voltage rails (+/-300V and higher), and the higher performance models (usually from the Stax "DIY" community) can push out a ton of heat too. In short, you gotta keep an eye on them. If your amp has push/pull balance control and DC offset trim pots, learn to check/adjust them. Keep a spare set of fresh tubes on hand - check/suspect the ones you have now.
If you had the 007 (or even 009) headphones I’d say that a Stax 007 amp isn’t going to be powerful enough to drive them to higher volumes without congestion (not unlike what you describe), but the Lambdas are generally much easier to drive than Omega series.
Also maybe think about whether your table setup & phono stage may be sending excess subsonic energy to the Stax amp, possibly overloading it?
If you had the 007 (or even 009) headphones I’d say that a Stax 007 amp isn’t going to be powerful enough to drive them to higher volumes without congestion (not unlike what you describe), but the Lambdas are generally much easier to drive than Omega series.
Also maybe think about whether your table setup & phono stage may be sending excess subsonic energy to the Stax amp, possibly overloading it?

