More tread does not necessarily make you safer on the road- if you think about a road tyre, like yours, on the road there’s a lot of rubber in contact with the road. If you think about a big knobbly mountain bike tyre then the tips of the knobbles will be in contact with the tarmac, which is much less rubber and so much less grip.
Off road a mud tyre, like a Specialized Storm, would work, but on road they will be more of a liability.The tyres you have already are probably the best for the job you are doing and the pressure sounds OK too. I think you just need to be a bit more careful if you’re not that confident about bike handling in the wet.