You are way overthinking this STD thing if you are leaving your boxers on during sex, or you have anxiety serious enough that maybe you shouldn't be having sex with people you don't know well.
First, you wore a condom. This protects your urethral opening, which means you aren't at risk for HIV, chlamydia, gonorrhea, NGU, mycoplasma and ureaplasma. The germs that cause these have to get into your urethra. Since you wore a condom, the head of your penis was covered, so you're fine.
Herpes, syphilis and HPV require direct skin to skin contact, either mouth to genital, genital to genital, or genital to anal. This is probably why you wore boxers. No STD can go through clothing, and none of these are transmitted by vaginal fluids.
You have eczema and psoriasis, and you had an open abrasion on your penis. I understand this causes more concern. You should avoid sex when you have symptoms, unless you know for sure your partner is negative for STDs, as broken skin can increase your risk for herpes and syphilis.
If your partner has a fungal infection, like yeast, having broken or irritated skin can increase your chances of you getting that, and getting a fungal infection on top of psoriasis and eczema is something you really don't want. Of course, this could happen with a long-term, regular partner, as many women get yeast infections, but a long-term regular partner is more likely to communicate with you about it.
Don't worry about keeping your boxers on after. That won't give you an STD.