I feel this, completely. To a degree it's why I haven't even been on the server at all lately; on top of being swamped with schoolwork/rl and not having the consecutive hours to spare, I feel like I can't engage at the level I'd like to. And when I do try to play, and play as my characters-- because that's what I do miss-- I feel guilty for not being as active as moderation staff as I'd like to. So either way I lose; if I can even find time to spend, I'm wasting it, either by not playing, or by not moderating. It's not a feeling I like, really, and I regret that I can't volunteer the hours I'd like to.
And as for pushing away friends, or being pushed away by friends, I also understand that feeling too. Getting invested in other people's characters has always been what makes me get invested in their players; if I can't play, or can't play in an immersed way, I have a hard time connecting.
I dunno if there's a solution, other than maybe, mandatory admin vacations or the like for active modmin staff, more delegation/dividing of tasks & who's responsible for them. It wouldn't change much for me, but more active staff, maybe. But actually enacting something like that is the hard part.