MAF14,
You say you're taking the T3 sublingually by placing it between your gum and lip. You may want to try taking it orally instead.
I have been taking Nature-throid under my tongue for over 10 years, and based on my over-range FT3 test results, I know I'm absorbing it when I take it sublingually, because I have no thyroid function otherwise. I think taking it that way ensures that the tablet is fully dissolved and also helps minimize the time before I can eat again. I know many people felt like oral Armour thyroid didn't work for them anymore after Armour changed the formula some years back, because of the microcellulose content. So I'm sympathetic to taking thyroid sublingually and believe that way is effective for dessicated thyroid products.
However, I have read that Synthroid and other levothyroxine products require stomach acid for activation, and therefore must be swallowed to be effective. Since Cytomel and other T3 products are synthetic, perhaps they need to be swallowed as well. If you've had success from taking Cytomel sublingually before, it's possible the formulation has been changed since then. Since sublingual Cytomel doesn't seem to be working for you, I don't see that you have anything to lose by trying a different way.
Edit: You probably already know that certain types of supplements, such as calcium and iron, block thyroid action and should be taken 2-4 hours after thyroid. You might consider whether you're taking any supplements, medications, or foods that might be blocking thyroid.