It's a very mammal-thing to do!
There are 2 versions: Submissive urination (a few drops to show submissiveness to the creature in front of you, don't eat me/you're boss!) & piss-scared (adrenaline dump shuts off digestion & evacuates the products therein. Generally in the following order: urine, vomit, excrement. It's why even if you remain in control, people pee more often when nervous. From job interviews to playing Hide'n'go'seek.) It can really be in almost any order, though. Round here you'll see a lot of people talking about anxiety levels being so high they're puking all the time, or have the shits. Incontinence of various forms gets less press, but I suspect it's just as common as the other 2.
This sounds like submissive urination, since it's just a little...Drives a lot of dog owners absolutely insane, that's where you'll find most of the literature pointed at if you google it... but all/most mammals do it. It's linked to lower levels of anxiety / wanting to be accepted rather than straight up terror. (More "you're boss" Less "don't eat me"). Baby mammals do it to/for nearly all authority figures. Adults it's less common. Per se. A lot of adults get a little turned on in front of people they seriously respect/want to like them. In certain bio psych circles this is explained away as actually a few drops of urine trying to escape, and the sphincter tightening to control it. Creating a little bit of chub, or a prolonged keigle hold. Shrug. Who knows? Maybe power is sexy, maybe we've all just got to pee.
Yes. It goes away, whether it's piss scared or submissive behavior. For the same reason; 'familiarity breeds contempt'. You become less afraid or more sure of yourself. In the meantime I suggest a pad. If you're male, a discreet one are "nursing pads". Which are slightly concave round pads filled with dry gel meant to slip into the bra of a nursing mother. Slips inside a boxer brief or tighty-whitey just fine.