Cost of ammo is important in the long run...
If you are budget limited at $350... and find $400 a stretch... you will have issues keeping a .380 fed enough for any real range time. I find the .380 costs about an extra $5 or more per box than 9mm.
I second the Sig 2022, and the CZ P-07. The Sig will push up about $100 more out the door, and the CZ will be about $50 more out the door than your $350 price point. The CZ you may need to get online to get the best price. I have seen the Sig going for $400 before tax at local shops, same as online.
Another is the Ruger SR9 or SR9c... both about the same price as the Sig and CZ. Really good pistols for the money.
You really gain a lot by adding the $50-100 to your budget. (keep in mind, the Sig comes with only one magazine)
Being as you may be recoil sensitive... and you say your main uses and want is a plinker on the cheap...
I also say a good 22lr pistol is a good choice, as they can be found cheap, and can be feed for relatively very cheap. (500 rounds for around $20) A Ruger Mark III, 22/45, or SR22 can be had for under $350 easy. (the Mark III and 22/45 can be more for the high end models)
The Mark III and 22/45 are very accurate and very mild recoil due to the weight. Getting a bull barrel version (large diameter, non-tapered barrel) adds to this effect.
I would suggest the SR22 as a really good pistol for your purposes. Good accuracy, low recoil, reliable, well built, less than $350.
Its not very large, so CCW could be done easily if you so choose... though 22lr is not the best defense caliber, it beats your fists. Use quality high/hyper velocity ammo round nose for best effect. I would use CCI Velocitors. They penetrate 16 inches in gel tests out of the SR22, from what I have seen.
After you get practice, trigger time and training... you can look into getting something more suitable for defense and CCW. Hopefully by then, you will have been able to better understand what you really want/need and can have saved up some more cash to make it happen.