Yes they would
as to the existing mails..
You' have to set IMAP up as a new account on each machine (using the same user name/password), and then stop the old accounts diownloading new mails
Mail already in the OLD accounts would not be synchronised unless you moved it into the new IMAP folders.
Here the thing to do would be to set one machine up as IMAP, move the old mails into it to synchornise upwards, then setup the second machine, let that synchronise downwards, then add any mails from that to the mix to synchonise back up again. Tedious but it works
Anything you don't want to synch, simply leave in the old account folders
The tricky part is what to do with new mails
Best bet would be to copy anything you only want stored on one machine into a local folder, and then delete the original
However that very much defeats the purpose of IMAP (it also probably explains your recent problems if you have the "keep on server" flag set)
Far better to use unique e-mail addresses for the two purposes