I am shopping for a shotgun, and the 3" shotguns are a good amount more than the 2.75" in the used arena.  Is the extra length worth it for range, killing power etc?  Looking to use for coyotes.

More is always Better!!

3" gives you the option of

(A) more velocity (range) for your regular payload weight

or

(B) more payload (pattern) for your regular velocity

Win or Win...

BIGGER_HAMMER

I like 3" for steel loads as there is enough volume in the hull to get a decent load.  The actual SAAMI pressure specs for 3" is exactly the same as 2-3/4" loads.

I've never played with lead buckshot or slugs, it could well be that they fit better in a 3" hull and you can get more pellets without exceeding pressure.

If you are a good shot it makes little difference.
It does not effectively increase your range any more than 23/4". You can change the payload and increase the velocity which may extend your range a bit. You can do the same thing with 23/4. You will just have fewer pellets in a given pattern.
Think of it like this: with a 3" shell you increased the amount of shot. In order for the increased payload to travel the same distance as the comparable 23/4" shell you have to increase the boom to push it. All you realy effectively do is fill the pattern with more shot.
So.. If you are a good shot you can use a tighter choke with 23/4" and achieve the same results.

Now. Having said that if you constantly shoot heavier shot sizes like 2,4 or BB then 3" makes more sence because you increase the number of large pellets so you do not have as many holes in the pattern at longer ranges. For upland game and the twice a year Duck hunt... 23/4 will work just fine.

As a practical matter. You buy the 3". Because you have more options in the future and you can still shoot 23/4" if you don't like the recoil of 3".

One minor practical matter.   If we are hunting upland bird (generally lead is allowed unless you're on a wildlife refuge or most of the tribal areas) I use 2-3/4".  For all steel loads I use 3" or 3-1/2".  It keeps it a little easier to not get caught with the the wrong damn ammo in your pocket.  You'll get a citation if you have lead shot loads on your person if you're hunting waterfowl - I know if I have a 2-3/4" shell in my pocket it's lead.  Even if the print is worn off.

Right now or planning in the future, clay, dove, quail, raccoon, coyote.  No place for waterfowl, and no buck fever.  Maybe freezer does, but shouldn't effect much.

 

No one every completely knows what the future may bring or what you may decide to hunt...

I have a pair of 3.5" magnums (870 & Benelli Nova) that I can shoot everything from the easiest of lite skeet loads, or stout goose getters...

I do hunt birds, so being able to use the same gun from Doves to Ducks, Quail to Goose helps me shoot more consistantly...