I have been hearing that grinding cross chaining sound when on my biggest chainring and my highest gear (smallest cog), which should not be happening. I tried a derailleur adjustment today and I'm wondering if something is wrong with the Derailleur or if I wasn't tuning it properly.

I have Shimano 105 components, here are the steps I took (FYI Low screw is closest to the frame, high is farthest - for you other 105'ers let me know if that's not correct, although I'm 99% sure it is):

-I shifted to my lowest gear and my small chainring
-I loosened the cable tension and made the low adjustment so the derailleur is ~2mm away from the chain.
-I retightened the cable tension, then I shifted to the highest gear/biggest chainring (the one that's been giving me problems)
-I tried to make the high adjustment but I can only see a visible move of the derailleur in one direction. When I try to loosen the screw the derailleur doesn't move outwards, it only moves in (when I tighten)...

To troubleshoot I loosened the cable tension while in high gear/big chainring and I get the same result.. I still can't get the derailleur out far enough to create a gap between the chain...

I'd rather not spend the money at a bike shop if this is something I can remedy myself so I thought I'd check here first.

It sounds like you have too much slack in your cable so the shift lever is unable to pull the front derailleur far enough to clear the chain when you're in your big chainring/small cog combo. I recommend putting your bike on the small chainring and big sprocket, then pulling the cable tight from there. Once you've got the cable tight you can use the limit screws and barrel adjuster to fine tune the shifting.

This worked. Thanks!

My front derailleur cable pulled through a loose clamp on a ride yesterday and needs fixing, so I'll try this too.

This may be geezer talk, but back in the day, you were supposed to avoid that combination because of the strain it put on the chain. I don't think I've ever used it.
On the other hand, the rules may have changed...I probably heard that 40 years ago and never questioned it.

I only suggested that gear combination because it requires the least amount of cable tension so it makes the initial tightening of the cable easier, regardless of whether or not you ever actually plan on using it.