I think that with the measures taken you are on a good way to recovery.
Dont worry about the dose increase for your SSRI, it helps you to stabilize and thats what you need right now. You can taper it down later when you figured out the rest.
Regarding T3; no, donf start on T3. I strongly recommend to start on T4 and to monitor TSH. There are good reasons why no medical association recommends T3 only treatment for hypothyroidism. T4 T3 combo is an option if T4 doesnt work.
With regard to T3 rT3 i highly recommend to watch this video
Take care!