Sounds like hypothyroidism to me. I have that. This can lead to dangerous things like not being able to think clearly at all, etc. etc
Bring baby with you but consider if you have any alternatives to holding the baby so much. We are talking about a short blood test. If your doctor usually makes you wait a long time, ask the appointment person and nurse if you can be seen quickly since you have a fussy baby with you.
You won't get the results right away. If you do have hypothyroidism, see a endocrinologist to get you started on the right meds and amounts and to understand all the what ifs and don'ts (no soy, no skipping or taking too much meds.. Then you can go back to a general doctor to continue meds and annual blood tests.
If you are not hypothyroid, ask the doctor what does he check next?
BTW, I never heard of hair falling out in clumps. Thinning a lot, yes. Clumps, ..... not really.
Does the baby have colic? If the baby's need to be held isn't medical, then you are causing this 24 hour dependancy. Not good for you or baby. Hugs.