Try a multivitamin maybe it will cover what you are deficient in.

Agreed, and I can't remember exactly and haven't got time to check, but I think my blood didn't normalise for eight years! For years I read this forum thinking why can't I get my numbers down - everyone else manages to.   But my gastroenterologist didn't seem to worry about it, which makes me think he either thought I wasn't complying to the diet, or he'd seen similar cases.

Yes, being off gluten for 3 months would likely yield negative results. To get accurate testing redone you would need to restart gluten consumption for several weeks (the "gluten challenge") to the tune of at least 10g of gluten daily (about the amount found in 4-6 slices of wheat bread).

That is one of the tests covered in the article I linked you above.

Antibodies to Deamidated gliadin peptides.  It is another celiac antibody test. The main test is the one you had, TTG,  But a full panel will also include DGP IgA and IgG.  I was positive on all of them!