Three miles is a very short commute, you'll be fine. You will be anxious on the roads at first, but your anxiety will quickly subside and be replaced by confidence.
Personally, I'd suggest buying a used bike. You will get so much more for your money. If you tell us where you live and what size you need I bet some people would even scout out rides on ebay and Craig's List in your area and suggest some good used bikes for sale--I've seen it done before, we like to get people on bikes.
If you want to buy new, $300 wil get you an entry-level bike. Nothing wrong with that, it will work fine and the shop you buy it at may include some tune-ups with the purchase of the bike. Many shops in my area include lifetime tune-ups.
For a commuter bike I would make sure that you can attach a rear rack as well as fenders. Anything that you can pedal that meets the above two criteria will work. You're only going three miles! If you're in good shape it will soon (if not immediately) feel like it is just around the corner.
I noticed you said you valued a "stable" bike, don't think that a tire with skinnier tires is inherently less stable that one with fat tires. Some novices are convinced of this error--not saying you are, just that many are. The fact is that if you are riding on pavement, a good commuting tire size could be as skinny as 28mm, or as "fat" as 37mm. Anything less is for racing, anything more is for off-road. I would add that control and stability are primarily to be found in the rider, not the bike. You will obtain both by riding, no bike can give them to you.
Any hybrid will do, though I wouldn't shy away from a higher quality used road, hybrid, or touring bike.
The above-mentioned Trek 7200 would do. I would prefer the 7200 FX. Watch out for the big, cushy seats on hybrids! They feel comfortable in the store, they feel comfortable when you press your finger into them. They are a sham! Cushy means more of you sinks into the seat, which means more of you touches the saddle, which means there is more friction created on your delicate body parts touching the saddle, you see where this is going and it isn't good. Just get a normal bicycle seat, not a barstool.
I'm not going to suggest more bikes, picking out a bike for yourself is part of the fun. If you find something you like, post them in this thread and I'm sure people will tell you what they think of them as potential commuters.