As Bercy confirmed, the tightening could be from friction (causing tearing & scarring). It could also be for a number of other reasons. In fact, there could be a connection between all your symptoms.
Eczema symptoms can be caused by yeast infections (not super common but it's possible--or at least they can co-occur as if relatedly). A lot of individuals have reported this, Google shows.* Also a yeast infection of the foreskin can cause it to tighten!
A better doctor might have suggested an antifungal of some sort originally. Hopefully it's cleared up already days ago? (Neutogena T-Gel is a dandruff shampoo that's also a really good over-the-counter antifungal soap for using anywhere.)
The foreskin has to easily fully retract or you risk it tearing during sex, and more importantly, easy retraction makes cleaning easy. If it has become too tight, it should be stretched open. This stretching can take weeks to months, don't try to rush it or it'll just take longer. And, as has been said (on this forum) already, "Don't tear the skin, that is the worst thing you can do, as it will heal up with scar tissue which is less elastic, making it even harder to pull the skin back. You need to stretch the skin, but do it gently so that you don't cause any damage to the skin." (to quote Zobo42--thanks, sir!)
A doctor with no knowledge of foreskin won't know to tell you this (though medical professionals really ought to know better), but basically, the only way to be sure to avoid the risk of infection in the future is to clean your penis better--and I mean a LOT better. In my experience, the only way to absolutely eliminate opportunities for infection (as well as any/all funky smells) is to keep my penis semi-obsessively clean. A penis with foreskin should be cleaned within a few hours after masturbating or sex by at least briefly washing with soap under running water for basically at least a minute. One should never go to sleep with any residue under his foreskin or in his urethra (the latter may be applicable to circumcised men as well). Festering gunk can cause an infection. Urinating to flush out gunk has some cleansing power (for the urethra), but it's not really enough to clean the foreskin IMO. It's better than nothing, but it takes a real wash to really clean the skin.
Basically my points can be summed up by saying: If you take pride in your hygiene, you can have an intact penis to be proud of. (Conversely, if you choose to neglect hygiene badly for long enough, you'll likely need to be circumcised eventually.)