I have a couple friends who have been working with a recent hypothyroid diagnosis. The first challenge is to get your treatment nailed. As mentioned above- T4 tends to be the first step go-to treatment, but anecdotally it seems, as dr. jim was suggesting, that a combo T3/T4 treatment has better results than just T4 (Synthroid). This is something that you need to have two things in play to work thru: 1) a doctor who is open to trying different medications & doses; 2) patience in letting each combination work to determine if it is accurate. There are no absolute or quicky answers to this because you are working w/ medication to find the dosing that is right for YOUR body chemistry. Not always the "average person's' body chemistry.
RE: weight gain - I would start w/ yourself first instead of a general worry about thyroid medication making you gain weight. I've had friends who have had challenges w/ dropping weight / bodyfat because they were hypothyroid, and at least if they are really honest about solid and real attempts to produce a body composition change (i.e a competition prep and not "oh I tried dieting for a few days and nothing happened, so I gave up becaues I must just have bad genes"). But in your cases, your height & weight, IMO are near normal and you didn't call out any problems w/ weight before the diagnosis. But if you are "skinny fat" now and want to change that, what are you doing now? The last thing you want to do is start any body transformation w/ the idea that you're unable to do anything because you are diagnosed w/ hypothyroid issues. That doesnt' immediately mean you're a victim of your metabolism and just generally ****ed.
Can you give a history of what is your current diet & training protocol? Your body is still only going to be able to respond to the environment that you give it to operate in. (i.e. your "lifestyle") If your diet isn't already optimized or reasonably clean or you don't get regular and consistent good exercise and good sleep, that alone will produce a great outcome of "skinnyfat". Typically you can make progress w/ small tweaks to whatever is your current lifestyle. Can you list your typical day's meal plan, e.g:
Meal 1: 8 am
1 whole egg + 2 egg whites
1 piece of whole wheat toast
1 cup coffee
Meal 2: 11 am
4 oz chicken
2 c brococli
etc. - whatever it is - the point is to do a real and honest look at what you normally eat, and then put that into a food counts program like
www.fitday.com (free, easy to use) to get a real and honest baseline of the breakdown of macronutrients (%, grams of fat / carb / protein), and total calories. People frequently are not able to accurately estimate how much they really consume in a given day so a reality check is extremely enlightening.
And also training / activity - how long training, typical week's training split, other activities.
And quality of recovery - how well do you typically sleep, how much do you sleep, are you frequently stressed out by work / life / family
All of this stuff, on its own can produce a 'skinny fat' result, regardless of thyroid performance.
I would start optimizing your lifestyle because this is what will promote and support and maintain a body composition change. The thyroid issue is somethign that you should continue to educate yourself on as well as continue working with your doctor if you feel that your current medication is not optimized for you. I think you should see other symptoms than just fat gain or not if your thyroid medication isn't doing the job yet - including things like mood swings. Particularly in women, the body's hormones are so tightly interdependent in producing the outcome of how well you feel and how well your overall body performs its daily activities.