Welcome to our forum, and am sorry you aren't feeling so good at present.
I always think if someone has had their thyroid gland knocked out or removed it can be a bit more problematic.
I am not too sure what your first doctor said about the two types of blood tests but it was probably only the TSH and T4.
As you have been on levothyroxine for about ten years now (I assume) and you are having clinical symptoms you need a Full Thyroid Function Test (GP may not do all of them but you can get those he doesn;t do privately and we have recommended labs).
You can tell GP you've taken advice from the NHS Choices Thyroiduk.org.uk for info on dysfunctions of the thyroid gland and because you are having clinical symptoms you would like to have:-
TSH, T4, T3, Free T4, Free T3, Thyroid Antibodies, B12, Vit D, iron, ferritin and folate.
Tick off the symptoms you have.
The blood test should be at the earliest possible, fasting (you can drink water) and allow a gap of 24 hours approx between last dose and test and take afterwards. This may prevent doctor adjusting dose unnecessarily.
Always get a print-out with the ranges and you can post for responses from members.