How to stop iPhone from "erasing its contents" when I want to "manually manage" it?

The solution is a two part solution.

If you routinely want to sync one device with two libraries, the easiest manner is to disable all automatic syncing on the first iTunes library before you bring the iOS device to the second computer.

You can always wait for the "erase" warning and cancel the sync. (which is the point you have in your screenshot). At that point, go into the current device settings and turn off automatic sync for all of the categories that cause the warning dialog to appear. Keep in mind, that music, ringtones, and other settings can all get the warning to pop up, so your work may not be done by simply un-checking music. The settings on the computer you get the warning are in conflict with the settings from the previous computer so you have to reconcile that before the manual sync can start.

Once you have figured out all the "silos" that matter to you, you can easily take one device to multiple computers for backup but only certain computers for syncing. Using one computer to sync the data (contacts and calendar) and another for the media is a use case that Apple fully supports in iTunes. Once you set things up, you should not expect to have issues with a one device and many computers situation.

Take a look at the answer to this Apple.SE question - it appears to have the answer to your question. You can try to transfer your purchases to the new iTunes account, since you mention that your other iTunes library is gone. Apps and ebooks should be transferable, but you may want to back up your music collection (if any) and handle that separately.

I have found the solution for it.

1. First of all download iExplorer on your Mac(Totally free software available)
2. Install it- You can click on "Continue With Demo"(so no need to purchase full version of it).
3. Now, connect your iPhone/iPod/iPad, iExplorer will detect it.
4. Now open Media folder, and you will see all your music files and data, now export it to any folder(please make new folder) whichever you like. It will start copying it to that folder - So your backup is ready now. (the names of all files are not as you are expecting).
5. Now, You are ready with your backup, so you can Erase and Sync your itunes, and can copy this backup again to your itunes without any risk.

Note: Sometimes, in step 4, Apple detects that you are backing it up, so it will delete everything in your folder on mac as well as from your iDevice. If this thing happens to you, then there is another workaround to solve it. In that case, open terminal on your mac and write this command in it, find ./ -name *.mp3 this will find all mp3 files on your mac(change mp3 to other file format whichever you want). Now, here you will find that files hidden in folder where you taken backup or in trash. Once you find that location, then right click on finder-> go to Folder -> paste that link over there, and you will be moved to some hidden folders, copy that files to somewhere else.