If you have test ridden the LHT, and it's your dream bike, go for it! It's tour ready and is a proven workhorse amongst tourists.
I own a 2016 Fuji Touring after looking at several options (including the LHT which was my
#1 choice). The Fuji was far cheaper, a better ride to me, and was clearly a capable tourer as well. It cannot hit the tire clearance of the LHT but does have friction mode on the shifters. For the money saved, I was able to get fenders and panniers to complete the bike. I am still left with about 300-400 I would've spent on an LHT that I saved for any repairs or parts swaps down the road. This is really kinda lame, but I don't like decals on bikes the same way I don't like branding on t-shirts and such so that also swayed me.
I don't know much about the AWOL - sorry.
All 3, like most any bike, can be toured on. Including the bike you own. Test ride, look at cost, and have a blast! I'd be more than happy to answer any specific questions about the Fuji.