Thanks for the help guys. I see Nashbars got their own 6-speed freewheel and they've got Shimanos, too. I'm not really worried about finding parts for this older bike, since they're not all that rare YET, and I'm not giving up 27" wheels until I can't find the $10 tires anymore! Besides, I like the way they ride.
I think what's going on is this: I've been taking training rides around the hilliest sections of town fully loaded plus a 50 pound bag of gravel in my trailer. I think the freewheel must be tightening on down since it's never had that much force applied to it before. Like I said, it hasn't slipped, so much as had a "mushy" feeling when I'm on really steep hills. If the chain had slipped on those hills I'd probably have wrecked from all the force I was exerting on the climbs! If my freewheel shatters into a million pieces, I'll let you know! Thanks again for the help.