Can I fix the hinges without replacing the screen?

it IS possible. Unfortunately there doesn't seem to be a repair guide for that specific problem, but you can follow the guide for the 15" Powerbook. It won't be exactly the same, but it should be pretty darned close. Check out this link:

PowerBook G4 Aluminum 15" 1-1.5 GHz Inverter/Antenna Cables Replacement

This is for the inverter, but the hinges are right there once you get to the inverter. I'm not saying that it will be easy.... in fact getting it back together can be pretty tricky, but it is doable. I've actually done it myself on a 15" powerbook.

the part you need to replace is the hinge "clutch" - it should be possible to do, if you're comfortable disassembling your machine. There are 2 clutches, left and right.

I've successfully replaced the hinge clutch on my previous laptop, a PowerBook G3 "Pismo". it required pretty well taking the whole unit apart - however repairing the hinges does not require you to replace the whole display. After taking apart the machine, it was clear which clutch was broken (in my case, only 1 was broken and needed replacing)

I bought the replacement clutch parts from another site several years ago (before I came across ifixit.com!) - but I notice you can buy these kind of parts here on ifixit too

I don't know the detailed inner construction of the Aluminum PB G4, but I think it's similar, as far as the display hinge goes.

the actual replacement of the hinge parts is easy - it's the disassembling the rest of the machine to get at them that's all the work.

good luck!