I’ve been shaving for over 30 years and have had problems since I was a teen, first from very severe acne and then from extremely dry and sensitive skin that was left over from the Accutane. In my 20’s I tried every new razor, shave gel etc going before I settled into a routine that works for me.
You could try looking at a double edge razor OK so they look a little old fashioned and it takes a little getting used to but I find they are much much less irritating than modern multi-blade razors. I personally find that the more blades a razor has the worse it leaves my skin. (As a bonus a pack of 10 double edge blades will cost you less than a single replacement cartridge for your current razor).
When you’re in the shower make sure you wash your face before you shave as this helps to soften up your beard and so you’ll need less force to shave. Less force means less irritation.
High quality shaving cream that you need to work into a lather and apply with a shaving brush also provides much better protection for the skin than the stuff that comes out of a can (even the expensive stuff that comes out of can). I use Geo Trumpers Rose, Violet or coconut oil. You could also try applying some shaving oil before you apply the shaving cream as this can help the blade run smoother.
If you need to shave against the grain then start by shaving with the grain then reapply shaving cream and shave against the grain. Don’t pass the razor over any area of skin that doesn’t have lather on. If you need to make another pass to apply more lather. At the moment I’ve been using a mainly a straight razor or double edge (depending on how much time I’ve got) and I can get a very close shave in 2 passes both going with the grain.
When you finished shaving rinse your face off with cold water and then gently pat dry (be carful not to rub) and then I always apply a moisturiser. I avoid applying aftershave as rubbing alcohol into freshly cut skin will always leave it red and sore.