Welcome to this forum.
For someone who is new to handguns, my first advise would be to do your research on the Internet. You can find a lot of information about handguns in general, handguns specifically for carrying and/or home defense, and actual reviews. Then, once you have decided on a choice through that research, go to a gun range that rents handguns and try them out. Rental rates are typically $20-$25 per range visit (for range fee and rental fee), plus ammo (for, may be, $15-$20 per 50-round box). Your choice will likely change. Then, buy the gun you want and practice with it regularly.
In general, when buying a gun, the price tag should not be THE deciding factor. After all, what good is a handgun if it malfunctions after 50 rounds or doesn't work at all with the self defense ammo you are able to buy. A cheap gun can be a nightmare and even dangerous if it fails to work the moment it is needed. A quality handgun will outlive its owner and can be found at a reasonable price.
For the .380 Cobra, there are a number of reviews out there, such as
Gun Review: Cobra CA380 - The Cheapest Handgun in America!. This article points out that the
Taurus 738 TCP costs just a little bit more, but is an overall better handgun in that price range. I have the Taurus 738 TCP as a jogging gun and I am happy with it for that purpose. Otherwise, I regularly carry a Glock 26 (9mm, 10+1 rounds) with an extra 12-round magazine. I carry a Glock 20 (10mm, 15+1 rounds) with two extra 15-round magazines when hiking. The Cobra CA380 review also mentions that its recoil is significantly harder than that of the Taurus 738 TCP.
Speaking of recoil, have you ever shot a 380 mouse gun, such as the Cobra CA380 or Taurus 738 TCP? For someone who is new to handguns, this is definitely NOT a handgun type that I would recommend. The perceived recoil is often described as "kicking like a mule." This goes back to my opening statement about trying guns out at the range. Small guns are easily concealable and look "cute," but they can have a quite stout recoil. One big mistake new shooters make is to get a gun that recoils too heavy, which causes shooting sessions to be less fun or no fun at all. A heavy recoil also results in flinching (jerking the gun/trigger before the bang), which results in NYPD accuracy (everyone/everything gets hit but the intended target).
Since the budget seems to be a concern, keep in mind that you need some target practice ammo, some self defense ammo, a holster, and practice after you bought the gun. For a new shooter, you should go to the range regularly and practice with your gun using cheap target practice ammo. Just buying a gun doesn't make you any safer. Knowing how to use it does. You also need to make sure that your self defense ammo cycles through the gun. Many recommend shooting 50+ rounds of self defense ammo through your gun. It should function flawlessly. A holster is needed as well, unless you want to join those few people every year that shoot themselves in the leg while carrying a gun without a holster.
My recommendation is to look at larger, quality handguns in .380 ACP or 9mm, such as the Bersa Thunder .380 (.380 ACP), the Glock 42 (.380 ACP) or the Glock 19 (9mm). If budget is really a concern, look for used handguns with known reliability, such as a used Glock 19 Gen 3.
PS: I know a woman of your height and weight that carries a
1911 Range Officer .45ACP Pistol daily. A good belt and holster is key for comfortably carrying a gun. Training and practice are key for comfortably shooting a gun. Purse carry is a good way to get your gun stolen and/or used against you.
A few resources: