I used both a diaphragm and a cervical cup (not together) for contraception back in the day, and the diaphragm (not the cup) for menstruation. It was great. Maybe 5 years ago I got a Diva cup and I just never really liked the thing (also how I felt about my cervical cup, which is a similar design). Hard to insert, hard to remove, then one day I dropped it down a porta-potty by mistake, so that was pretty much that.
Then I discovered the Instead cup, which is a LOT more like a diaphragm, and that's my current system. They're supposedly "disposable," but I haven't disposed of one yet, just wash with soap and water same as I did with all those other tools, and reuse (I've never boiled any of them and doubt the Instead would survive it). And count me among the fans of the "take it out in the shower" method. Sure, it's got a bit of a Psycho look to it, but it's by far the easiest context to deal with the mess, IMHO (porta-potties, OTOH, are at the low end of this scale, even ignoring the dropping it problem).
For public restrooms: wipe out thoroughly with toilet paper, wrap, and stash in your purse, using another clean one until you get to a private location for washing the cup with soap and water. A nice thing about Instead is that you get a 6-pack for not much more than $6 (see e.g. Amazon), so you can keep one stashed in your
car bicycle, one in your purse, one in your office, and yes, carry a spare on you when you're out and about. Though I find I rarely need to change mine other than a.m. / p.m.