Wow, some great suggestions so far.
I also suggest a Dozier. His D2 is about as sharp as sharp can get and it holds an edge for a very long time. It's also easy to sharpen back to scary.
After more than 3 decades of hunting and trying more knives, styles, and steels than I can easily recall I've settled into using a Chris Reeve Nyala in S35VN.
I'm able to do multiple deer on a single sharpening. It takes a VERY keen edge, is pretty easy to touch up with a ceramic rod in the field if need be, and is stainless- which is a nice thing to have in Michigan weather.
The only downside in my opinion is the excellent leather sheath that it comes with. I despise loop sheaths that I have to feed my belt through and I almost never take a leather sheath into the woods. I prefer Kydex with a Tek-Lok for the gear that I actually use, easy on, secure, and easy off.
Kydex can be sourced for a small cost that'll still keep the pair under your budget. The handle is long enough with enough traction to keep it solid in your hands without being too aggressive. The front of the handles are scalloped to aid in using a pinch grip, a grip that I use a lot when field dressing, the choil is deep enough to keep you away from the edge and it's chamfered and rounded for comfort.
It's a very stout blade from thick stick that has a hollow grind so that it slices well and is not fat behind the edge.
ESEE makes great knives that come with great sheaths for way under your budget!
The carbon steel stays sharp for a really long time and they have a top tier warranty in case you ever really mess it up. I just found that using coated blades is something that I don't like when it comes time to clean the goo off the blade after a successful hunt.
I believe they make uncoated stainless now which would be worth checking into.
An issue noted by many, including myself, is that the handles are kinda short on the older numbered styles they make. Plenty useable, just a bit short. There are aftermarket handles that fix the problem and I'm pretty sure the newer models " named instead of numbered " have longer handles. Just some food for thought.
Good luck on your hunt for both blades and game