I replaced the nozzle on my S1 Pro with a Tungsten nozzle, but the issue is that the nozzle is too high. I think the nozzle I bought is a tad shorter than the stock nozzles. The CR touch sensor touches the bed before the nozzle does. The bed is level, and the gantry isn't quite square (different issue I need to fix), but the left side is higher than the right so that should have helped me in this situation. Is it possible to lower the heat break to bring the nozzle down a millimeter or so?
You should first fix that the gantry is straight and level (not bubble level, but square with respect to X, Y and Z).
In order to adjust for a shorter nozzle when using a touch sensor you need to re-calibrate the Z-offset. See e.g. Automatic Bed Leveling (ABL) with a sensor (BLTouch, inductive, capacitive), how does it work?. With a shorter nozzle, your offset will increase. You shouldn't lower the heatbreak, this may have adverse effects in feeding the filament into the heatbreak.
The CR touch sensor touches the bed before the nozzle does.
The nozzle isn't too high. The sensor is too low. Unless you mean the probe touches the bed first while in extended state which I doubt you mean because where's the problem in that? - that's how the probe is supposed to work!
First check how the probe is mounted on its mounting bracket. If it's mounted on its underside, shift it to its top side. If it's already on top, either add some washers/distancers to the probe mounting screws or adjust the 3D project of the bracket and reprint it. Afterwards perform standard Z probe offset calibration.