The limited test results you have are pretty awful. Not only should you push for Free T3 testing, you should start pushing for T3 in your med, either from just adding T3 med, or by going back to Armour. I say that because it is unlikely that T4 only is going to get your Free T3 level high enough to relieve symptoms. If your new doctor is one that refuses to prescribe T3 type meds, you need to find out about that now.
Obviously you need to get your Free T4 level to the middle of its range, at minimum. You also need to test for Free T3 so you will know how much that needs to be increased. Also you need to find out if the new doctor is willing to treat clinically, by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results, and especially not TSH when taking thyroid med. If the new doctor is unwilling to treat clinically and prescribe T3 meds then you will ultimately need to find a good thyroid doctor that will do so.
If you want confirmation of what I say, have a look at the following link. I recommend reading at least the first two pages and more if you want to get into the discussion and scientific evidence supporting all that is recommended.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
Hypothyroid patients are also frequently deficient in Vitamin D, B12 and ferritin. It is important that you get those tested and then supplement as needed to optimize. D should be at least 50, B12 in the upper end of its range, and ferritin should be at least 70, and some sources say 100.
I meant to concur that high cholesterol can be related to hypothyroidism. Also, be aware that the conversion chart you used is most likely wrong. Most of them show that one grain of Armour is equal to 100 mcg of T4. That is incorrect. If you look at Recommendation 6 on page 11 of the link above you will see the scientific evidence that one grain of desiccated thyroid med like Armour is equal to only about 66 mcg of T4.
Due to the difference in half-life of T3 compared to T4, by stopping the Armour and going to the 150 mcg of T4 your symptoms may be a bit worse for a few days until the increase in T4 med starts to kick in.
You absolutely need to get tested for Free T3. But even if tested now and confirmed as very low, I doubt that the doctor would be willing to consider adding T3 to your med until after being on the 150 of T4 for 5-6 weeks. At that time you should make sure they test for Free T4, Free T3, Reverse T3, cortisol, B12 and ferritin. All are very important for a hypothyroid patient to know. In the link you will also find that B12 should be in the upper end of its range, so you could start supplementing with B12 now. I also expect that you will find your ferritin level too low. It should be at least 70 and some sources say 100.
As you proceed with this doctor keep in mind that a good thyroid doctor will treat a hypothyroid patient clinically, by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results. So if it were me I would ask the doctor if she is willing to treat clinically, as described, and also if willing to prescribe T3 type meds like Armour and Cytomel. If either answer is no, then you are going to have to change her mind, perhaps by giving her a copy of the entire link, or else you will need to find a good thyroid doctor.