I think you have a good plan.
I completely understand your frustration.
Please know that you will get better.....the reason things are taking longer than they should was due to the fact that you weren't being medicated properly.
Your new Synthroid dose is still on the low side and, unfortunately, you will need to wait 6 wks. after every dose adjustment before adjusting the dose again (if needed).
If you change your dose any sooner, your body will never have the opportunity to truly adjust to a dose and you might overshoot your goal. In the meantime, you'll be increasing the intensity of moving thyroid hormone levels - this is what causes symptoms as well.
So, while your meds doses have been doubled, you still aren't at your optimal dose(s).
But, your doses are being increased as they need to be (that wasn't happening before).
You will reap the benefits of your patience and hard work.
I can assure you that, once you reach your optimal levels, there will come a day when you wake up....and you'll realize the "old" you is back!
It happened to me enough times in the past to give me strength to bear with future dose adjustments and now, I'm finally enjoying some extended time of feeling awesome.
Just keep on getting labs every 6 wks., push for the meds dose adjustments that your labs indicate and the same will happen to you.