I've been riding my cross-check for about a year and a half now, and I love it. However, there is one problem I have with it, and it's that the brake hoods aren't that comfortable for riding on. I feel like I'm overextended.

I've come to the conclusion that I either have one of two problems:
1) the stem is too long
2) the handlebars are a size too large for me

I did take my bike in for a fitting about a year ago, and they told me that the handlebars are a bit wide for my size, but I'm curious as to whether the wideness goes hand-in-hand with the depth of the handlebars.

Any thoughts?

Width and depth are unrelated, but obviously, the wider they are, the further you'll have to reach.

Try a shorter stem because that's an easier fix, and if that doesn't work, try new bars.

You can also move the levers on the bars or rotate the bars themselves to bring the hoods closer to you. I've rotated my bars up a bit in front, so the flat portion between where the bend starts and the top of the hoods is nearly horizontal. Grant Petersen recommends that on the Rivendell website, and it works for me.
According to folk legend about bike fit (which is often wrong, IMO, but never mind...), a good starting place is to aim the open end of the bars at the rear brake, then set the levers so their lower ends just touch a ruler held against the lowest part of the bars, pointing forward. Doesn't suit everybody, but it gives you a baseline from which you can adjust.

I've experienced a similar issue with bar width. Mine was that the bars were too narrow and it caused me shoulder and upper back issues. I do agree with your LBS, bars make a huge difference in fit just like other things. Bars are pretty inexpensive, maybe give the size that they suggested a try.