Congratulations on your new home!* Unfortunately it sounds like your peppers have phytophthora blight.* It often starts at the soil line, but can start at other places on the stem when a plant gets it and is older.* Take a look at these sites and pics.* Not all the sites have all the info or the best pics and that's why I've included more then one.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.