DCrom, where is your tongue when you play? I never thought much about it before, but tonight I noticed that if I draw it back a bit kinda like I’m saying “errr” I get a surprisingly more powerful tone and the overall pitch is a bit sharper. It makes my high notes more in tune (seemingly) without having to lip them up as I usually do.
What I’m doing is consistent with the advice from Nicholson’s book on Terry McGee’s website and Rob Greenway’s irish flute page. It’s worth a shot.
Ooops, on reading some of the comments following mine, I see I was vague … thank you for illuminating that. When I say lift your chin, I mean raise your whole head, almost as if someone’s said something mildly insulting to you and you’re responding with a “hmmph!” or a sniff. Try it a few times and you can see how it rolls the flute out slightly and drops it just a bit down your chin … generally you use it more as a quick fix for a flat note or passage, but I’m thinking if you’re pushing the flute into your jaw really hard and rolling in a lot it might be an interesting alternative position to try.
Yes, you can move your jaw around small amounts, but more up and down (to open your throat or what have you) as opposed to backward and forward – but that’s a place that’s best visited with a trained professional.
Jennie raises a great point again! which leads me to one other question … could it be that in trying to get a really solid low register on a slightly leaky flute you’ve overcompensated? Are your joints nice and solid? It might be interesting to try playing the flute in a fairly neutral position – i.e., level, with the blowhole basically aligned with the toneholes and the blowhole properly placed as outlined above (place flute edge in the middle of the lips, roll out, etc.), and not too much air – and see what kind of tone you get. If it’s kind of puny, you may have a bit of a leak and you may have fallen into the flat/hard-blowing habit to try to overcome it.
Anyway, hope that’s more clear. 
cat.
Thanks for the advice & encouragement, people. It helps both technique and motivation. 
A couple of developments to report:
Last night, I set up my Folk Flute with the embouchure directly in-line with the R2 & L2 holes, kissed the far side of the embouchure, and started to play. And - much to my surprise - found myself in-tune with the recorded tune I was working on. Still some problems hitting the top notes in the second octave reliably, but much less than I remembered. I think that a large part of my problem was that when I first started playing, it was easier to get a note with the head rolled in, and it became habit. It does seem bit harder to play with the head out, but it appears that my embouchure has strengthened enough that I can handle it. Just needed a (figurative) kick in the tail to get me motivated enough to try again.
Casey Burns got in touch with me to offer to trim the tenon if I needed it. Given the low price point that Casey sells the Folk Flute at, and that I bought mine second-hand, that’s going above and beyond on his part - kudos to Casey. But I thanked him and declined, telling him that any problems were due to my (lack of) embouchure rather than his design. I will send my FF in for a checkup one of these days, but not because it can’t play in tune.
Interesting. I recently became aware that my tounge is usually right up against my bottom teeth. I’ve started to make a conscious effort to pull it back a bit, and I do feel my tone is more solid.
That’s what I was doing beforehand. Now my high A and especially B, which had been flat before, come right up to pitch, and the tone is more in line with the lower notes, whereas it had been more hollow before.
I have a FF. I have to slide the headjoint out about a quarter of an inch most of the time to get it flat enough. I play with the headjoint rolled in and I blow down into the hole. Before I started doing this, I couldn’t get it flat enough even with the joint out as far as possible.
PS I still find the low D is a bit flat compared to the other notes.
Odds are, the low D is flat on purpose to allow you to blow a hard D.
Eric