I think it's probably just personal style, I don't much like FR myself.
I think if you're fancying a change to 6max just suck up some losses at first if necessary - see it as a training expense. As long as you're reviewing hand histories, learning new stuff each session and so on I doubt you'll spend long as a losing player at 6max 10NL if you consistently beat FR at that limit. If you've only played a little 6max yet anyway, it could just be variance.
As far as adapting your game goes, well I only play 6max so I guess I can't say from experience, but it seems to me that just as M2M said, when UTG, UTG+1 and UTG+2 fold at a FR table, you're now playing 6max, so it's not as though it should really be any different.
Another thing that you didn't mention is how many tables you play - if it's more than about 4, it would surely help to drop back to like 2 or 3 tables of 6max at first, then add more as you get comfortable with it.