Sounds like your 20ga had a bad combination of shot size/ounce and choke size, but to say 20 gauges do not pattern well is just simply wrong. All shotguns, no matter what gauge, should be patterned on paper until the proper shot size/ounce works with the choke of the gun for the average distance you will be shooting.
Also, to let everyone know, the problem with locating 16ga ammo is because small stores can not afford to keep ammo in inventory that is not common to most hunters and target shooters. Small stores may try to keep some amounts on hand, but what happens is when a 16ga shooter arrives at the store he will buy it all for himself, leaving other 16ga shooters dry and empty. Sorry, but that's a fact.
So if you are shooting 16ga, the best thing to do is order enough cases of it to last you all the hunting seasons of your life. Or take up reloading.