How should I handle cracked grout in my two bathrooms?
Just caulk it. Grout will tend to crack in any change of plane, so caulk is the recommended practice. For prep, scratch out anything that's loose or protrudes past your caulk profile. You'll probably want to use white for the tub transition and a color matched caulk for the tile-to-tile corner.

A typical installation is caulked at the plane change and has room for both planes or at least the floor to expand comfortably. Grouting it with no expansion room is what you have. The "right" thing to do would be to scrape it out to give it more wiggle, but it's not absolutely necessary. It's a lot more work and possibly more likely to break your tile compared to just leaving it be, especially since it's been fine for a year so far.