Is there a way to clean or flush old galvanized plumbing lines without damaging the pipe?

In short, no.
Best to make a game-plan to replace them - you can do so in sections to budget the cost and time to do it.

Pipes are being "flushed" every time you open water. As suggested above, best way to deal with corroded pipes is to replace them. Once rust sets in, there is really no way to remove it.

Hope that helps

Milo

I agree with Milo and RickJ, Thanks.

John

You don't have a choice until you replace all your corroded pipes, make a good decision and prevent big pain.

Your next and only step is to retrofit all the corroded piping. May sound expensive, but it really isn't . Find out what size piping and how many feet approximately are corroded. Your best bet should be replacing the piping with PEX tubing, it's cheap, effective, and will not corrode.