There are in the latest Shimano catalogue these 44-32-22 cranks:
- FC-M371 - square taper, riveted chainrings, 9-speed
- FC-MT101 - ditto
- FC-MT210-3, riveted, HT2 9-speed
- FC-T4060 - removable chainrings, HT2 9-speed
- FC-T4010 - removable, Octalink 9-sped
- FC-T3010 - square taper, removable chainrings, 9-speed
In general, you might well want to replace your BB, and if you're doing so the best option would probably be to go HT2, in that square taper is fine but Shimano are making progressively lower quality (cheaper) square taper BBs, and quality third party BBs will be expensive. Octalink is dead and will be combined to the trash can of history soon enough, so ignore that as you don't already have it.
It's very cheap and easy to remove your crankset, a crank puller is like $3 from China.
or $5 for a slightly fancier one with a handle
Unfortunately removing the bottom bracket can be more challenging and you will want to buy a branded tool. However if your bottom bracket does need replacement then it's not significantly more expensive to replace it with a HT2 model.
This is the tool to remove an Octalink or Square Taper BB
Whereas a cheap $5 tool like this will do for HT2