It might. If it doesn't, a lot of people use an aggressive razor on their face, then a mild razor for their dome. Decent DE razors like Qshave, and Weishi are cheap. Even this $6 Wilkinson Sword razor https://www.amazon.com/dp/B0012Y1FCK might help a lot.

Try one of these Wilkinson Sword Classic razors including 5 blades for $5.98 and see how it compares to cartridges. https://www.amazon.com/dp/B0012Y1FCK

An alum block can help a lot with irritation. It's not a cure, but usually gets rid of a lot of it. You could also try this $6 safety razor. https://www.amazon.com/dp/B0012Y1FCK A lot of people end up going to them to get rid of ingrowns and irritation.