We have had members with Hashi's report having a rash also. Here is a link to one thread on that subject. If you want to read further, you can use the search button and find others.
http://www.medhelp.org/posts/Thyroid-Disorders/rash/show/960923
You are really missing the most important thyroid hormone test, which is Free T3. Free T3 largely regulates metabolism and many other body functions. Scientific studies have shown that Free T3 correlated best with hypo symptoms, while Free T4 and TSH did not correlate. Members taking a T4 med often find that their body does not convert the T4 to T3 adequately. In view of its importance, you need to request testing for Free T3, along with Free T4 each time you go in for testing. If the doctor resists, then you should insist on it and don't take no for an answer.
As noted in the link above, that member found the rash to get better with an increase in meds. I expect that you will find your Free T3 to be too low in the range, consistent with still being hypo. Are you having any other of the 26 typical hypo symptoms listed in this link?
http://endocrine-system.emedtv.com/hypothyroidism/hypothyroidism-symptoms-and-signs.html
A good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free T3 and Free T4 as necessary to relieve symptoms, without being constrained by resultant TSH levels. Symptom relief should be all important, not just test results, and especially not TSH, when taking thyroid meds. You can get some good insight into clinical treatment, like you will need, by reading this letter written by a good thyroid doctor. Whenever the doctor consults with a patient from a distance, the letter is sent to the PCP to help guide treatment.
http://hormonerestoration.com/files/ThyroidPMD.pdf
In the letter note the statement, "the ultimate criterion for dose adjustment must always be the clinical response of the patient."
So I think you should go back and get the Free T3 test done, along with Free T4. Since hypo patients are frequently low in other areas, it would be good to also test for Vitamin D, B12, ferritin, and a full iron test panel. With your symptoms and the low dose of T4, you should discuss an increase in your meds, with your doctor.
When test results are available if you will post results and reference ranges shown on the lab report, members will be glad to help interpret and advise further.