I recently started shaving with a safety razor (the next step to a straight razor from my old Mach 3). I usually shave twice, the first time with the grain, and the second time against it. When I shave the second time around my chin, main of the hairs bleed. I don't believe that these are ingrown hairs because there is no swelling later on. The skin is not bumpy, the hairs are just bleeding from where they were taken out. The bleeding is not from cuts (I'm decent enough at shaving to avoid slices). I should note that my hair is fairly thick.
Can someone recommend a way to avoid this bleeding? Currently, I just put some petroleum jelly on it and I am fine afterwards, but it is a hastle. Should I just not shave against the grain?