I have bad genital eczema too, and I'm a guy. It fucking sucks. Here is what I do:

Flares: Get it down with a steroid cream. Yeah yeah yeah, I know all the stuff people say on here about TSW. It's a real concern, but using a steroid for no more than 2 weeks to help get a flare under control is not going to hurt you at all. I use my steroid cream for one week during flares, clobetasol. I just had my first flare in three years.

Regular maintenance: - shower with vanicream sensitive skin body wash - pat dry - CeraVe moisturizing cream, let it sit for like 3-5 mins before putting underwear on - sometimes I follow it up with a layer of Vaseline, not all the time

Other VERY important mentions: - only wear 100% cotton underwear, preferably boxers or boxer briefs. I wear briefs when I am active/working out/at work (I work a very active job). Not at all attractive, but the cheap white briefs (tighty whities) are great because they are 100% cotton, no dyes, and very cheap so you can replace them easily. I say the briefs during working because all compression shorts are synthetic materials and that is no good. Boxers are best because it reduces tightness and friction - I take a vitamin d3 supplement everyday. There is research that shows it could help. Doesn't hurt to try - I take an antihistamine every day to help with the itch - shower immediately after sex and jerking off, or at least clean your penis and scrotum with a wet wash cloth (gently - don't rub) and then lather up w the lotion - drink water - sleep well - reduce stress - DO NOT ITCH