So by no means do I intend to become a vegetarian, but maybe a pseudo one. I seem to have some symtoms of hormone imbalance and cannot take anything to correct this due to a blood clotting disorder. So I am looking to my food and feeling maybe a few meat free/low dairy days could benefit me.

From that perspective, what would be more beneficial - one week a month vegetarian on a few days week?

Are you planning to cut meat down on your 'non-vegetarian' days? I would say few days a week - that way you keep your diet more consistent on average, and its easier to gradually cut meat out every so often than go veggie for a whole week. Perhaps try both ideas out and see which one feels better

So I've tried it, almost accidentally, a couple times and the entire day has been meat free. I'd like to take it a step farther and have less to no dairy, but I'm taking it slow. When I can I always try to by non-hormone treated dairy and meat, but I'm on a budget and things of that nature can be difficult to find.

After further thought, doing a 2-3xs week sounds like a more balanced, sustainable approach.

In the UK its pretty easy (and not much more expensive) to get organic meat, things are probably different in America though. If you do manage to find some non-hormone meat, perhaps buy a little amount, freeze it and use it sparingly in dishes? that way your getting the benefits of a mainly plant based diet, with a little bit of meat too. I think non-hormone treated usually means better welfare for the animal too, which is always good.

I agree that a few days a week sounds like a better approach.
I'm vegetarian plus I eat at least one Vegan (no animal products) meal a day and on Mondays I am fully vegan. After doing that for awhile, I found I liked the vegan diet and now 90% of my cooking is vegan, I pretty much only eat dairy when I go out.
So you could also try something like 1 vegetarian meal a day (or one carnivorous meal a day) depending on how much you would want to cut back. Check out the Meatless Monday website- lots of great meal ideas for non-vegetarians wanting to give meatless a try.