Hi, welcome to the board, I don't know much at all about hyper as I am hypo, no doubt someone more in the know will pop in and comment soon!. Your TSH of 16 does show hypo though, this was probably a result of the thyroid being suppressed to much on 20mg and your doc was right to reduce it. High antibodies means it is an autoimmune issue and possibly Graves' disease if you are hyper. However, hashimotos which is autoimmune hypo can swing between both, it is very early days in your treatment and can take a while to become stable.
You and your doctor will need to work together to find the right med level to control it. It is a good idea to read up on what you can so that you have an idea of what is going on. Doctors are well known for treating only TSH levels and once taking tablets for thyroid starts it is generally unreliable after that and thr FREE T4 and FT3 are a better guide to work from, though I am assuming it is the same for hyper as for hypo.
Do not take your tablets before your blood test as this will skew your results.
I hope someone more in the know for hyper treatment can help you out! And that you have a good doc who can manage this for you, but do find out as much as possible as you will need to be very active in the managment of it. It is very treatable but can sometimes be a bit of a long haul finding the right dose! You are in the right place for help here! Best wishes