With all those symptoms and your possible high TSH level you really do need to get the tests done, as recommended by both Red_Star and myself. If you want to be confident about this information, I highly recommend reading at least the first two pages of the following link, and more if you want to get into the discussion and scientific evidence for all that is recommended.
http://www.thyroiduk.org/tuk/TUK_PDFs/The%20Diagnosis%20and%20Treatment%20of%20Hypothyroidism%20%20August%202017%20%20Update.pdf
When you have test results, please post them along with reference ranges and we will be glad to help interpret and advise further.
Please tell us about the symptoms you have. Feeling better is not going to be an issue with accuracy of tests.
TSH only starts to rise late in the day, so anytime in the morning/early afternoon is okay.
The tests that you need are TSH, Free T4 and Free T3, not Total T3. Free T3 is the biologically active thyroid hormone metabolized in body tissues. Total T3 is not nearly as revealing as Free T3, so insist on Free T3. In addition with the prior high TSH you should also be tested for the possibility of primary hypothyroidism (Hashimoto's Thyroiditis). Two tests may be required: Thyroid Peroxidase antibodies (TPO ab) is the one that usually shows up. If it is negative, then the second test is Thyroglobulin antibodies (TG ab). In addition it would be good to test for Vitamin D, B12 and ferritin.
These tests should go a long way toward identifying any thyroid issues.: however, symptoms are the most important consideration in diagnosing possible hypothyroid ism. That is why I started by asking about your symptoms.
The August 2012 study entitled "Clinical Significance of TSH Circadian Variability in Patients with Hypothyroidism" (Sviridonova MA, et. al) showed an average morning TSH level in the subclinically hypothyroid group was 5.83 mU/L and in the afternoon was 3.79 mU/L; 73% TSH circadian variability.
The average morning TSH level in the hypothyroid group taking levothyroxine was 3.27 mU/L and in the afternoon was 2.18 mU/L; 64.7% TSH circadian variability. Morning: 8 - 9 am. Afternoon: 2 - 4pm.
Test your thyroid labs in the morning. I prefer to fast before testing. Labs to request include TSH, free T4, free T3, thyroid peroxidase antibodies (TPOAb), thyroglobulin antibodies (TgAb), vitamin D, iron studies including ferritin, vitamin B12.