A TSH of 2 is supposed to be perfect, an optimal TSH is a TSH of around 2.5 .
i've seen my TSH climb from 4 to 7 in that time period, with significant antibodies and one side is swollen. However, he didn't want to treat me yet which I found frustrating.
Some endos do not treat until your TSH is above 10, however, a TSH of 7 is quite high and need to be treated.
Since you are positive for hashimoto's disease, you need to repeat your TSH test in three months interval, once you start meds or a new dose, it reduces your TSH, but in three months interval, your TSH will rise again indicating the need to increase the dose, this process repeats itself until you reach the proper dose for you.
Dose should be calculated per week and not per day, this is because Levothyroxine has a half life time of 7 days, which means it reaches steady state in six weeks, increments are mostly of 50 - 100 mcgs per week.
It would be great if you can get your doctor to test for Free T3 and Free T4 in addition to the TSH, but according to my own personal experience TSH is sufficient in the vast majority of the cases.
And an important issue is that since you have a swollen one side, you need to perform thyroid and neck ultra sound to see what is in there, there might be nodule/nodules or goiter, Levo shrinks them in third of the cases, in an another third they remain the same size and if not cancerous are left and in third of the cases they grow and need to be removed by a surgery.
Best of luck,