I know what you mean about being overcommitted: on top of working and being married and having two children growing into and through their teens, Vix and I spend a lot of time on our shared avocation, which involves time with various portions of a big and diverse community.
Vix doesn't have a full-time job - she had been a homeschooling mom until last year, then spent time traveling and visiting Doc and developing her part in the avocation to something like part-time work; she's also looking to develop a tele-working kind of career for herself.
Anyway, Vix has had more time than I have for developing relationships. That once might have been An Issue for me, but I don't really fuss about it, now. She is mindful that I need time for myself and other relationships, even if that time is necessarily limited; I"m happy to give her the opportunity to travel and explore, since it meets many of her needs.
In answer to the more basic question, what I've come to is the realization that being poly doesn't really take time at all: it's more of an attitude toward the world and toward other people, a way of being in the world. In my case, it's also an unconventional way of understanding what my marriage to Vix means.
It's developing actual relationships that takes time and energy, both of which really are in limited supply. I've really scaled back my expectations of such relationships, enjoying them for what they are and what they can be under current conditions . . . though without placing arbitrary limits on what they might be!
Full-time "romantic" commitment isn't really in the cards, but all manner of degrees and kinds of intimacy may be possible short of that.
Just consider that other poly folk tend to be very busy, too!