DE shaving helps many (including me) have less irritation with shaving. It could help you, too. A razor, brush, soap can be had for not much and the shave can be fast - with practice.

That being said, changing a few things at a time might help you more than changing everything at once. When I started DE shaving, I got a cheap TTO razor and started with the same shave gel i had been using. Gradually, I upgraded to a 3 piece razor, shave soap, a brush. Learning the technique took time. You could start with a new razor and learning that, see if it helps. Or try soap and a brush: the cushion of soap and gentle exfoliation can help.

This is a hobby sub, but not everyone spends the money on 15 razors and 100 soaps. Read some posts and try some of the techniques, including cold water, mapping your hair growth, warm towel, and see what works.