Welcome to the forum, Malmeslady.
Palpitations is a hypothyroid sign but some people have TSH in the hundreds at diagnosis whereas others can barely function with TSH 4.0-5.0.
TSH >10 is overtly hypothyroid (high). You haven't included the range for FT4 so I can't be sure whether it is low or below range and I suspect it is the latter.
For maximum absorption Levothyroxine should be taken with water 1 hour before or 2 hours after food and drink, 2 hours away from other medication and supplements, and 4 hours away from calcium, iron, vitamin D supplements and oestrogen.
It takes 7-10 days for Levothyroxine to be absorbed before it starts working and it will take up to six weeks to feel the full impact of the dose.
You should have a follow up thyroid test 6-8 weeks after starting Levothyroxine. Arrange an early morning and fasting (water only) blood draw when TSH is highest, and take Levothyroxine after your blood draw.
Ask your GP to test thyroid antibodies at your next blood test. If antibodies are positive for autoimmune thyroid disease (Hashimoto's) it may be beneficial to adopt 100% gluten-free diet which can reduce the symptoms of Hashi's and antibodies. Otherwise eat whatever you like with the exception of unfermented soy which is very bad for the thyroid.
Most people will find symptoms resolve after their TSH drops to around 1.0 with FT4 in the upper range but symptoms can lag a couple of months behind good biochemistry.
High cholesterol was one of the diagnostic clinical signs of hypothyroidism prior to blood tests being introduced. As your thyroid levels improve your cholesterol will usually drop.
I am not a medical professional and this information is not intended to be a substitute for medical guidance from your own doctor. Please check with your personal physician before applying any of these suggestions.