Yes, your thyroid meds can affect your liver. You might want to discuss this with your endocrinologist or primary doctor, whomever is responsible for your thyroid management.
Your ALT enzymes are produced from your hepatocytes, so the issue involves your liver somehow, not else where. But given the relatively low levels of elevation and the normal imaging, then your liver is very likely still in good shape. It takes a very long, chronic disease to push your liver into an advanced disease state. Let's hope your new doctor comes up with something soon for you. A biopsy might be an option.
Your puzzle remains it seems, did your doctors offer any possibilities that they are considering? Also, have they tested your autoantibodies yet, given the ALT and fatigue then AI issues are a possibility. Clearly your ALT is elevated enough to indicate that there is something amiss.
Have your doctors mentioned a liver biopsy yet? When all else fails then they may have to consider doing this.