Don't worry about stopping power, worry about accuracy. A 22 short to the noggin is more effective than a 500 S&W that completely misses the threat. As far as the 380 many officers carried that round during WW2 on both sides. Soviet police carried the 9X18 for decades which is basically a 380 on steroids.
380 is what my wife mostly carries as she shoots that round with more confidence than a higher powered round such as the 40 S&W even though it was in a full size firearm. Her current carry is a Witness Pavona about the size of a Glock 19, she shoots the gun with confidence, and can easily hit paper plate targets(head shots) at 10 yards. She has also on occasion carried a NAA Black Widow 22lr in a neck holster while driving to Florida. Again she can put the rounds on a paper plate at 5 to 7 yards. The neck holster is one of the quickest ways to deploy a handgun in a car if needed. A 22 to the face is going to make a very bad day for a threat.
So it is not about power, though if you can handle, and shoot well a more powerful cartridge and deploy it effectively then that is probably the way to go. I would suggest see what guns you can borrow, or rent, then shoot them on small targets, the smaller the target the better. Or if using silhouette target train also for head shots. Consider, test holsters, and how you will carry as to which firearm you pick. You will find that eventually you will probably end up with more than one firearm.
There are some very good open carry poly firearm options that are affordable Witness, and SAR as well as a Smith SD9. The Smith, and SAR usually sell new for less than 300, a compact Witness is in the 350 range. If you just want to OC until you get your permit to CC then you can save a lot by buying a Hi Point, many scoff but they work, and are under 200. One of the first guns my wife bought was a CF380, that gun has over 1000 rounds without a single failure. Due to it's weight it is just like shooting 22 even though it is a 380. A HP C9, Fobus holster, 500 rounds of ammo would probably be less than 300 bucks.
If you go with the last option come back, and I will tell you how to break in the HP to have as good luck as my wife has had with hers. The same routine though should be followed with any new semi auto.