I used to have similar problems (actually, that was why i got into SE/DE shaving to begin with.....40+ years ago!). For me the problem was/is a curly beard. Shaving with twin/triple-blade razors would cut the hair below the skin line and when the hairs start to grow back, they would curl in causing irritated bumps.
My solution then was to switch to an injector (still have my Schick from college!), but DEs or Single Edge razors might be helpful for you, especially should you not wish to attempt straights (which i am also doing at this very time!).
You can beat these things! Best of luck.
"[FONT="Trebuchet MS"]A true gentleman is one who is never unintentionally rude."[/FONT]