A 20" barrel will kill a turkey no problem as long as you do your job when pulling the trigger.
But with the looser choke of your 20" (Likely a cylinder choke), your range will be more limited than using a longer barrel with a full choke, or even XXfull choke and will make it harder, so honestly, having a dedicated tight choke barrel wouldn't be a bad idea at all for turkeys if you're not going to be able to set up pretty close to where you think they're going to be. Around or within about 30 yards max is where I would be most comfortable with a cylinder choke barrel and #4 shot as an example.
Kind of the same answer with a slug barrel for deer, however, if you're hunting out of a blind or stand and within 50-70'ish yards or less, the barrel you already have, and a standard slug will piledrive deer all day and all night. Again, provided that you can put it where it needs to go, but that should go without saying.
So, I guess the short answer would be, if you can hunt fairly close, you could use what you already have, or if you're going to be taking much farther shots, you may want to consider a dedicated barrel accordingly.