My dermatologist recommended single use razors, since bad stuff can collect on razors between uses and cause skin irritation/rashes. Cerave is great for sensitive skin
My skin did the same with angry red bumps every time. I use the Venus razor with the moisturizing strips and put Aquaphor on my legs several times a week at night. The Aquaphor helped about 80% and I use Boiron arnica cream or gel after to soothe and calm the red bumps.
I have an all natural skin care company. I recommend our bomb butter. It is a whipped body butter made with only 3 ingredients (organic raw shea butter, raw organic coconut oil & essential oil). It works wonders for me.
6 years ago I switched to shaving using a safety razor with disposable blades. This totally stopped my irritated red bumps. Search for 'Long Handle Butterfly Open Double Edge Safety Razor.' I don't think brand is important but mine is WEISHI. I shave in the bath and don't use any shaving cream/lotion - this works for me using a safety razor but not 'regular' razor. I have to shave slowly and carefully to avoid nicks! However, not only is this better for my skin, it's also better for the environment and more cost effective.
You'll also need to buy razor blades. I bought Astra, but I'm not happy with my most recent pack so will try something else. I just read that Astra is an "aggressive" (i.e. sharp) blade so I'm going to order Viking "mild" blades. I'm guessing this will be easier to shave with since these blades are designed for (coarser) beard hair. I might also buy an adjustable safety razor.
When I'm in a hurry / in the shower rather than the bath, I use Billie razors and Aveeno unscented shaving cream.
For body lotion I recommend CeraVe unscented daily body lotion. Or if your bumps are persisting, something with glycolic, lactic, or salicyclic acid: Paula's Choice AHA body lotion, Cerave SA Lotion for Rough & Bumpy Skin, or AmLactin.
It's really important for sensitive or irritated skin to use products without any fragrance and without alcohol (note that things like cetearyl alcohol are not the same thing as regular alcohol and are moisturizing rather than drying as isopropyl alcohol and alcohol denat are). Most (but not all) products from CeraVe, Cetaphil, and Eucerin fit the bill. Pricier but good brands include SkinFix and Paula's Choice.
This is a much bigger investment than a product or new razor, but I got tired of dealing with this and eventually invested in laser hair removal. Seriously the best money I've ever spent. I've had my full leg, underarm and bikini area done and it's been life changing.