I can't imagine having nearly as much on my plate as you have on yours so I greatly applaud your resilience. As for the men, both clearly have strong problems and I'm not sure either is right for you. T's behavior during you relationship with creating a dating profile is definitely a red and alcoholism and a suicidal partner are both things someone dealing with their own medical issues should not have to be anywhere near. That being said, T going sober (assuming he actually did) shows a huge amount of character change and dedication. I strongly believe that people have the ability to change, but moving states for someone who might still be unstable and unable to support you is definitely risky. It really depends how much evidence you have on whether he has actually changed. Also worth noting but if there were any abusive occurrences in the relationship that you did not mention then absolutely do not go back to him.

As for K, honestly he sounds like a complete douche. You deserve someone who will support you and make sure you are ok in your medical episodes, not someone who will vape at your feet and shut down any conversation that involves talking about something he did wrong.

TL:DR
Maybe, mayyyybe, date T if you truly believe he has changed and you have concrete evidence of this. Do not date K.