I just got back into the game of flashing phones after being out for a tear it two, what us dirty flashing, and in on stock 5.1 just unlocked using sunshine and rooted and installed twrp.
Dirty flashing is flashing the rom without wiping any cache or data. You should go ahead and make a backup in twrp now. Also install titanium backup and then make a backup of your apps and data. Copy. Both twrp and titanium backups to your pc so you can get back to them if need to. The twrp backup is not in the directory you might expect. I'd tell you but I don't remember off hand. If you search these forums it won't take you long to find.
After that you can flash a rom. The one you linked earlier was to get to 5.1 from 4.4 via mofo. If you want that one go to First Encounters thread. It will tell you what to download and how to flash it with twrp. You could also try Computerfreeks.
I don't know of a way to dirty flash what you are on, and guarantee that that you won't lose your apps/data. If you have your titanium backups it will be easier to restore. Still a pain but easier. You'll download titanium again, then restore your apps. That's better than downloading all of your apps again. If you haven't used titanium before, you need the pro version and should practice freezing, backing up and restoring before you flash any rom's.