Yes, I've been hypoglycemic for about 8 years, I've had a thyroid problem for about 4 years. I have more problem with it when my thyroid is off, they sort of go hand in hand, some people tend to have higher blood sugar when their thyroid is off and some have lower.
Your FT4 is low and this is probably affecting you more. If you get a glucose tolerance test done be sure it is a 5 hour not a 3 hour, the 3 hour can sometimes miss the problem.
If you are hypoglycemic this means you need to stay away from sugar and processed foods, eat lots of vegetables especially green ones and avoid white flour products, whole grains are better. Be sure you get enough protien and eat often, small snacks every 2 to 3 hours.
First thing I recommend is to find the book "The Low Blood Sugar Handbook" by Ed and Patricia Krimmel, it is very informative and lists foods to eat and foods to avoid. Also it would be wise to get a glucose monitor and check your blood sugar when you feel poorly to see what level you are at. In time you will learn the signs of when you need food so you eat something before you get too large a drop.
Typically what they look for in the glucose testing is a drop of 30 or more points in your blood sugar within a short amount of time. The large drop is what causes symptoms even if its a "normal" number, its the amount it falls that makes you hypoglycemic. Mine can be 110 and 20 minutes later be in the 80's, the 25 to 30 point drop in such a short time causes me to feel off, sometimes dizzy, shakey etc. even though 80 or in the 80's is a "normal" level, its the drop thats not normal.
Hope this helps you some, keep us posted.