It can take a while to get a diagnosis and that may depend upon the doctor.
I would suggest a second opinion. I can't imagine going to a PCP with complaints of abnormal thyroid blood work and they do a hands off physical and don't even check your neck.
I would urge you to follow your instinct and get the second opinion or consultation with an endocrinologist you mentioned. Your elevated TSH level ought not be dismissed as easily as it seems your current doctor is doing.
If I understand correctly, she merely wants to retest the T4 in six weeks but didn't begin a treatment program, is that right? That approach really doesn't make sense to me. Clearly something is out of whack with your thyroid function, and it's not going to fix itself — it can't fix itself.
Here is a link to the WebMD overview of hypothyroidism. I hope the information is helpful for you as you continue to approach this matter with either your current doctor or a new one.
Good luck, and let us know how you're doing.
Welcome to our board.
Your doctor is "on the ball" by ordering the thyroid panel.