Do you know how LARGE a too hot signal would have to be? Your typical SS amp has 15v rails. To exceed those, your signal would have to peak OVER 15v. That is over 10v RMS of signal. Can your tube screamer pedal produce 10v RMS? If it runs on a 9v battery, it only has about 4.5v headroom. That is 10v short of the 15v cap. and only about 3vRMS of signal.
SO I really doubt you are even overdriving the input, at least in terms of signal level and clipping. Can you be hitting it with signal too large for it to amplify it well? Sure, why not. But that won't damage it, it will just clip the signal.
Now what if I did put 16v of signal into that IC running on 15v? Most modern designs have clamping diodes on the inputs, and for that matter, even on the outputs.
DO you have a selection of PV schematics? A few examples common enough you may already have: XR600B, Combo 300, Bandit (most any version), Heritage VTX, KB300. That is enough, you have any of those? Pick one and look at the input circuits. From the input line you will see a pair of diodes, one to +15 and the other to -15. Those are normally reverse biased to they are not conducting, and are invisible to the audio. If some voltage hits the input and goes above 15v (either polarity), then the diodes conduct and shunt the excess voltage into the low impedance of the power supply. That prevents the input pin of the op amp from being more positive than its power pin. if the input were allowed to exceed the power supply voltage, the IC would think it had reversed power supply. Damage could result.
Now look at a power map like the PV 400BH. You will see reverse biased diodes from the output to either power rail. They do the same thing. Not concerned the amp will drive the output past its rails, but the inductive kick of a speaker sure could.
If your modeling amp does funny things, since most of the amp takes place in the digital world, I'd be more thinking there is some gating or other level control inside the program. ICs have no memory. Well, not unless they are memory chips.