I found one study that says TSH increases in response to domperidone. "The rate of increase varies with the functional state of the thyroid, as confirmed by results obtained in cases of euthyroidism, primary hypothyroidism, subclinical hypothyroidism and subclinical hyperthyroidism."
https://www.ncbi.nlm.nih.gov/pubmed/4094840
That said... Is TSH the only thing that's being tested, in relation to your thyroid? Your doctor should be ordering Free T4 and Free T3, as well, every time you have a TSH test.
Free T4 and Free T3 are the actual thyroid hormones, whereas TSH is a pituitary hormone that should not be used as the sole basis for determining thyroid hormone status.
How long have you had Hashimoto's? It's not unusual, in earlier stages of Hashimoto's to swing back and forth, from periods of hypo to hyper or normal. Also, if one is on thyroid hormone medication, dosages, often, have to be higher during a pregnancy, then decreased once the baby is born, since the mother must provide thyroid hormone for, both, herself and her baby.
The other thing is many medications can interfere with thyroid medications. Most medications shouldn't be taken within an hour or 2 of taking the Synthroid, so it's possible that the domperidone is preventing the Synthroid from being fully absorbed, also. I did read, as well, that domperidone may also cause weight gain, as does being hypothyroid.
Are you making sure you're staying hydrated, since you stopped the dairy? Your own dehydration could have something to do with not having enough milk as well. From what I've read, it's not recommended that you just stop the domperidone as it can cause severe side effects - it should be weaned from, but I can't find specifics, right off hand, about weaning or how long it should take... you should discuss this with your doctor.
If you have results for Free T4 and Free T3, we'd be better able to help with your thyroid issues, if you'd post those here, along with the reference ranges, which vary from lab to lab and have to come from your own reports. If you've not had those tested, you should ask your doctor to order them as soon as possible so you know exactly what's going on, since a TSH of 33 is quite high, as you know.