I also have a couple of Baofeng HTs and little budget to work with, and I have 2 go-to antennas and a couple others I use on occasion.
The one I would recommend is a Comet SMA(in my case, BNC with an adaptor)-24. Performance on 2 meters is similar to a plain 1/4 wave vertical I made out of a bit of wire and a BNC connector, but the Comet's much less awkward than my homebrew and seems more durable. Its performance on 440 is nothing to write home about, but is still significantly better than the factory duck or the 2m 1/4 wave.
I have no personal experience with the Nagoya antennas mentioned here by others, but from what I've read there are quite a few knockoffs floating around Amazon and eBay, so buyer beware. If it costs less than $20-30, it's almost certainly a knockoff. That being said, I have a knockoff Harvest RH-770, which has a similar Nagoya equivalent (NL-770), and it performs even better on 2m than the Comet or the homebrew 1/4 wave, but it's only so-so on 440 and it's awkward and delicate so I'd avoid it as an only antenna because when it inevitably breaks, you're in trouble.
Though it's not directly applicable to your requirements because it's not handheld, I'll mention my other go-to antenna, an N9TAX dual-band roll-up slim jim with 16' of coax. This one I hang in a tree or from whatever I can find, as it's happiest with at least 6 feet of free space around it. The higher the better, as it has a low take-off angle, and the results with it 20' in a tree are nothing short of astounding compared to any handheld antenna, both on 2m and on 440. It rolls up small enough to fit in a coat pocket.
Each of these can be had for about \$30 and are available with an SMA connector to directly fit your Baofeng radio, but if you want to use more than one I'd advise getting the BNC versions and spend another \$5 for an SMA/BNC adapter so they can be swapped quickly and easily without too much wear and tear to the radio's SMA connector. All 3 of them and an adaptor is still under your \$100 budget, and gives you options for most portable (Comet), most gain (N9TAX), and compromise (RH-770) situations.