Joshua, I have a question. By what you said about Onegaishimasu, is it correct to say that it has a similar meaning of Doozo?
Just "Josh", onegai shimasu.
Doozo and onegai shimasu are quite different, meaning-wise.
One use of "doozo" is an intensifier in requests and prayers. In this way it might seem similar to "onegai shimasu", since it's so often related to "negau", humbly requesting or praying for something. But it's merely in an intensifier, essentially meaning "somehow, in some way".
"Doozo" is used in this manner, then, when offering someone something. Imagine you offer someone some food. In Japanese society, you downplay the quality of whatever you are giving someone, so you say, "Doozo omeshiagari kudasai". Translated somewhat stilted, "Please partake of this somehow." From this kind of usage it's become now commonplace in any kind of situation where you offer or give something to another person, or indeed, just handing someone something.
Me: So-and-so-san, could you hand me that stapler?
So-and-so-san (grabbing the stapler and holding it out in my direction): Hai, doozo.
Me: A, doomo.
And so now, when one says "Doozo tabete kudasai" (please eat this), it doesn't so much have the sense of "somehow, in some way", but rather, "Go ahead and eat, please!"