Sounds like you have symptoms of hypothyroidism. Lab tests often miss this so it is best to test the low tech way. Look up how to do your basal body temperature. This is conjunction with symptoms is the most accurate way of determining if hypothyroidism is an issue.
Other symptoms that may occur include:
Weight gain, feeling cold, constipation, dry hair and skin and hair falling out.
The dizzy/shaky feeling sounds like a hypoglycemic reaction, which can occur with poor adrenal function. Poor adrenal function also leads to thyroid suppression. Therefore, you should also look in to low adrenal symptoms and support.