My stove looks similar. More important than looks, it has the spark ignition for the burners and oven.
"I went through the manual and I saw where it said you could adjust oxygen being sent to the burners (by sliding some metal covers that allow more/less oxygen in) but I didn't see, and there were no pieces provided, to change the burner orifices themselves."
The Premier site states that all stoves are sent out set for NG, but the swap out orifices for LP are supposed to be shipped with the stove. Sounds like a glitch in service if you did not get any.
For the top burners, there is a screw behind the oven knob that also must be moved from the NG to the LP setting. There is supposed to be a small hole in the metal behind the knob through which you can put a skinny screwdriver to turn that screw. My experience was that on the first stove (damaged in shipping) that the hole did not line up with the screw...on the second stove, they forgot to make the hole...so I had to remove the metal panel to access it. If your burners are sooty, but not your oven, maybe this step was missed?
The "metal covers" you refer to I think are only one, and that is for the oven. It is all the way in the back, in the center bottom of the oven, behind the broiler drawer. The "cover" is a vertical cylinder on a metal gas tube, and slides from left to right around its axis. Below that cover is a nut, which must be turned all the way the "other" way from its original setting for NG.
Each one of the burner orifices for the stove top are different sizes. I think it is important to carefully match each orifice with the proper burner, as each burner is already set to feed different flows of gas through the orifice.
I got in touch with Premier on the shipment damaged first stove and found their customer service was pretty good. I was in contact with a man who had obviously been there many years and knew the stoves inside out. They shipped me a new one, and although not a perfect solution, under the circumstances, things turned out well. I encourage you to phone them, soon. I like cooking with propane, but have a deep respect (fear? caution?) about it's volatility.