that faucet adapter is meant for a party-type system where you can keep the keg in a cooler or trash barrel of ice and just tap right at the corny keg post and not have to worry about a line or a tower or a cooler box.... problem is, right at the post there's not usually enough resistance to keep the beer from just releasing all it's stored CO2 right as it comes out. you need a way to simulate a longer beer line which adds resistance and keeps the CO2 in suspension...
two easy ways to do it -
search this forum for mcmaster carr mix-stir sticks - they are these spiral-type plastic epoxy mixing sticks used to mix a 2 part epoxy as it comes out of the tube... but if you just the large end off of them, they are the right size to fit INSIDE the keg's liquid-out dip tube - you may need 2-3 of them inside the dip tube to create the sort of resistance you'd get out of 8-10ft of 3/16th inch beer line.
another option is to fabricate a riser out of piping to raise that tap to chest height if the keg is sitting on the ground - you'll sometimes see these things at beer festivals and stuff. looks like this...
this too will create extra resistance which will cut down on the foaming issues.
as to carbing, best results are when you carb and serve at the same pressure - dialing back and forth is an inexact science and can mess with the results.
regarding your 5lb cylinder, a 5lb cylinder should last VERY long even when used for carbing and serving. like 8-10 kegs worth at least. if you're blowing through one tank per keg then you def have a leak somewhere in the system, use a starsan solution and spray it on all your line connections, the quick disconnects, the posts on the keg, the lid of the keg, the pressure release, etc.