Is there a way of keeping the songs on my iPhone without having to keep them on my computer?

if you turn off auto sync you have to manually delete songs rather than the computer doing it automatically.

go out and buy you a external hard drive.

can you set it to manually manage music in iTunes? I have an iPod touch(i live in Canada) so I’m MOT sure if you can do that on your iPhone.
also, if you jailbroke it, you can install swaptunes, so it doesn’t update your iPhone every time. But I’m not sure if it will work, cause its supposed to be so you can sync music from two computers. Good luck though!