Can I T my new dryer's exhaust into my bathrooms exhaust?
You do not want to T it into your nath exhaust.
You do not state if the dryer is gas or electric but because it is 120 volt, I would think it is gas. If it is, the results of you suggestion could be lethal. Do NOT do it.
If it is an electric dryer, you still have the problem of all the moisture going where you may not want it. Remember, air flows to the least resistance and if for some reason that is back into your bathromm, or from the bath to the dryer, then that is where it will go, along with the lint, moisture, and whatever else it carries with it.
Aside from this, I suspect it is against building codes to do this.