My droid started acting up several months ago. It started turning off randomly at first then it wouldn't turn on at all. I went to take the battery out and to my surprise the battery had become swollen or something. I got a new battery and within a few weeks it started turning off randomly as well! Then it wouldn't survive off of the charger. And then, it wouldn't turn on at all! SO annoyed! Since I am at home most of the day I leave my wifi on and use my home internet instead of my data. I read somewhere that when the Wifi is used too much it starts having problems. But I don't understand why it would do that!
I'd say that the second battery that you bought is bad as well. A swollen battery is definitely a bad thing - dangerous as well, as it suggests that the battery pack could combust.
I have no idea where you read that about using WiFi. I use WiFi all of the time and my phone is better, with better battery life, with WiFi connectivity than with mobile data.
At this point I would not buy a battery for the D3 from anybody but Motorola, unless you know it is in the retail pack. A lot of the cheap Amazon batteries are, I think, used batteries. Moto still sells the extended battery, but you need to buy the door as well.
Battery BF6X (Extra Capacity) - Motorola Mobility LLC. USA
Extra Capacity Battery Door for DROID 3 - Motorola Mobility LLC. USA
It's expensive, but it should be a full capacity battery, and will give you about 20% more battery than the standard. If you are not doing so already, I'd also suggest using the stock Motorola charger and cable, just to make sure that a cheap charger isn't overcharging the battery and causing all of those problems you've had.
Of course, if you have a D3 you may be close to upgrade time. It may be better to use that money on a new handset.