Having spent last summer living in a black mold infested rental house, I am extremely suspicious you have a mold problem in that house, probably a very toxic one, and should focus your energy on that before looking into other causes.
Your symptoms are mold-exposure symptoms to a T. I thought it was just a coincidence or moving stress that I got a cold the day after we moved in. It really wasn't, and the ongoing health problems only get worse. Anyway, you can do your research online about the symptoms of black mold exposure if you want, I'm sure you'll be like "wow, yeah, I didn't realize I was feeling that symptom too, but now that you mention it..."
First, I would contact a lawyer to discuss your situation. There's a good chance you are going to need one. Leases are legal documents. And quite frankly, the kinds of landlords who ignore water leak problems long enough for black mold to develop are often the kind who are unwilling to actually fix the problem adequately, and/or willing to hire sleazy contractors who will lie and say they see no evidence of mold, so there's a good chance of needing to take the landlord to court and/or break your lease. The lawyer we contacted gave us free advice before we moved out about what sort of evidence to gather and so on, and then took our case on contingency, so it didn't cost us anything up front. Where we live, undisclosed mold is a legal reason to vacate without notice--but that's why you might want to talk to a lawyer to make sure that is the case in your area as well, but the legal issues are probably outside the scope of this site ;-)
Secondly, get some mold testing done. We found the home test kit we got at home depot worked accurately. However, if you end up having to go to court over the mold issue, having a professional mold inspector do their own testing on your behalf is helpful, because they can give their unbiased testimony in court, whereas you certainly are biased and they can't prove or verify you actually tested where you said you did. In particular, I would focus your test kits on places near water--under the sinks, bathroom floors, anywhere where there's stains or discoloration on the walls or flooring as well as places where the floor seems soft, soggy, warped, etc. Your lawyer may be able to advise a reputable testing company should you run into problems finding one willing to test a rental property (we had issues with that...). If you aren't confident that there is a mold problem, you could do the inexpensive home test kit, and then decide whether or not to seek more expensive professional testing based on the results of the home test kit.
Third, get out! As fast as you can. Don't wait around for the landlord to fix the problem. It probably goes deeper than you realize, and will take more time and effort to fix than you're going to want to put up with.
In our case, the shower was leaking. And the kitchen sink. The other sinks and tubs may or may not have had leaks in the past, but they weren't leaking when we moved in. It takes multiple months of continuous water issues to go from harmless molds to the kind that release toxins into the air. The first sign we saw of a problem was stains on the wood flooring in the room opposite the shower, and then a soggy spot on the floor. And the lower kitchen cabinets had a horrible odor that I couldn't identify the source of. (definitely look for any/all of those signs) But after everything unraveled and we had the professional mold tester come out and the previous tenant showed up at our doorstep while we loaded our moving van, we realized the mold, being airborne, has made it's way into the HVAC system, and anytime you turn on the heater or air-conditioner, you're blowing mold spores all over the house, the landlord had painted over the mold on the walls before we moved in to disguise it, etc. The wall facing the shower had mold "spider-webs" covering the entire sheetrock on the side hidden inside the wall. We certainly couldn't have seen that when we moved in, but when the landlord cut a hold in the wall to make sure the shower valves weren't "still" leaking, it became immediately apparent there was mold. Not only that, within two days we were all experiencing exposure symptoms severe enough we decided to immediately go stay in a hotel until we could find a new place to live. Should you decide to cut holes in the wall to look for mold (I don't recommend it), I would wait until after you move out, and wear a respirator and gloves.
to properly remove mold, any affected sheetrock, flooring, cabinets, or other organic material that can harbor spores need to be removed, as well as leaking pipes fixed and so on. But before doing so, a negative pressure system needs to be set up so that any mold that becomes air-borne is blown outdoors a fan instead of blowing into the rest of the house. Your landlord should NOT hire some random contractor to do the work, they should hire someone specifically experienced with mold-removal, preferably with certifications to that effect, mold is not something to mess around with, you don't want to leave part of the problem by mistake ;-).
If your landlord's homeowners insurance covers mold and/or water damage, there's a chance the whole removal and repair would be handled by the insurance company. But to be honest, I wouldn't stick around in a rental long enough to get the landlord to clean up the mold. If it's making you as sick as you describe, it's probably not going to be a trivial renovation needed to actually fix it to the point of habitability. Plus, medically, you are sensitized to mold now, and will be more bothered by any residual mold than someone without mold exposure history.
Decontaminating your belongings, even after you move out, is not a completely trivial matter either. Vacuum everything with a HEPA vacuum. Vacuum your floors every day if you can. Wash every single article of clothing, sheets, etc. Wash with soap and water all the kitchenware. unfinished wood furniture may require using an EPA registered mold-icide to kill or remove the hyphae (mold "roots") in the wood. Throw away your pillows, all of them, get new ones. Don't mess around with that one. While they are washable, the puffy interior can harbor spores and toxins and they don't always wash out in the washing machine...and you put your face on it for 8 hours a day. I'm sorry you're going through this, I wish you the best of luck.