TSH alone not much useful anyway do these blood tests from a private lab for avoiding any further delay in treatment/supplementing.
Vitamin D3. Vitamin B12 and Ferritin
If these not optimal can itself lead to hypothyroid and other fatigue symptoms.
Concerning thyroid do these basic ones Free T3, Free T4 and TSH.
Concerning thyroid hormone replacement there are mainly two types, levothyroxine and natural dessicated (NDT) one and also another like T3 only with mixing with levothyroxine etc. Levothyroxine is choose by most doctors but I think everyone should go on a trail on natural dessicated ones too at least for some 1 month and choose the one which gives them better feeling.
As des900 said, TSH, alone, is not adequate for determining actual thyroid hormone status. You need to have Free T4 and Free T3 tested, as well. Those are the actual thyroid hormones and correlate with symptoms, while TSH does not correlate.
Vitamin D, B-12 and iron deficiencies don't "cause" hypothyroidism; low thyroid hormone levels cause hypothyroidism. Deficiency in some vitamins/minerals can cause hypo-like symptoms, plus iron is necessary for the proper conversion of Free T4 to the usable Free T3. Ferritin is a storage hormone and indicates the amount of iron in storage.
If you can get Free T4 and Free T3 tested, please do so, then post the results, with reference ranges, so we can see your actual status.
Agree with des and Barb. You need to get all those tests done. Also, since you mention hives and your TSH level has increased it would be a good idea to also test for the antibodies related to Hashimoto's Thyroiditis. For that, two tests are required: Thyroid Peroxidase antibodies (TPO ab) and Thyroglobulin antibodies TG (ab).