In factory loads I've used #4 Buck 3" Mag and Hevi Shot T 3" mag DC loads out of my Pump, I don't remember ever taking any long shots, most have been in close do to the cover I'm in.
I've shot #2 Hevi Shot in my 20ga but again never had to try and stretch it out as I was shooting a combo gun with the other barrel 223 for anything that wasn't up close and personel.
I load 2 1/2" 1 oz nickle plated BB's in my 16 ga Drilling, haven't shot anything with it yet but it patterns good to 40 yrds and again I don't have to stretch it out as I have a rifle barrel for the longer stuff.
I've tried 2 3/4" #4 Buck in my 12ga combo gun and while it did the job a finisher shot was required both times, it never patterned that well at 40 yrds. I've handloaded a great 2 3/4" load of 33 pellets of nickel plated F shot and 1 1/4 oz of nickel plated BB's and they both pattern well at 40 yrds. I just loaded a batch of 2 3/4" 1 oz ITX BB's and patterned them yesterday at 40 yrds and they looked good. Again all of these are for combo guns so getting extra distance isn't a big deal.
So I guess for me in a shotgun only area it is #4 Buck in lead areas and Dead Coyote in lead free areas out of my pumpgun. Luckily I was in a store in WI and they were trying to get rid of their DC and I picked them up for $19/bx and took 5 bx so that should last me a long time as little as I use a shotgun only.
F-BB should be effective on coyotes and add 2's in fox areas. The only way you are going to know how they work it to get out and pattern them, every gun is different. Try different choke tube, the tightest isn't always the best.
Here is a long running post on shotgun chokes and patterning for predators.
http://www.predatormastersforums.com/forums/ubbthreads.php?ubb=showflat&Number=777351&page=1