Thanks for your responses, everyone! I'll have to check the parameters again when the fishies are awake- it's almost midnight here. No need to freak them out. The ammonia, nitrites, and nitrates were all at 0. I don't remember the pH at the moment, although I remember that it was in the "ideal" range. I also don't remember the number for the water hardness- that was my only concern. I think it said "moderately hard" on the test strip, but I've heard that cories like soft water. Other than hardness, everything else was at 0 or within ideal limits.
They seemed to be acting much more normal today. I did add 2 more, so we now are at 5, and they're pretty much hanging out together. (The 2 that I added were from the same store, same tank, only a couple of days apart.) They are finally eating well and eagerly as soon as I drop the food in. They're still moving quickly, but not darting back and forth as quickly as they were. They're more interested in the bottom of the tank now, but they'll still swim around and check things out all over. And they're not going to the top excessively for air, so all good news there. Overall, I think they now seem happy and healthy, even if they weren't before. They're hanging out together or at least grouping up sometimes, they're checking out the driftwood and the live plants, they're eating ravenously, etc.
The betta is next to them at the moment. I wanted to see how he would react before doing putting them together, plus quarantine anyways. He watched them a little when they first went in- not flaring or anything, just watching, but he's lost interest. There's definitely a plan B- he can go right back in his old tank if he's a bully!
He's currently in a 5 gallon, but I really wanted some cories. And I obviously didn't want to add cories to a 5 gallon tank, so I decided to upgrade the betta to a 10 gallon so I could get some cories!
It's been a little frustrating, though. I kept reading that a 10 gallon tank is fine for them (I live in a small apartment so I can't go much bigger. I also have several other bettas and a saltwater tank...), and that a minimum of 3 cories is fine. I asked the employee at my LFS when I bought them the other day how many would be a minimum number- I want them to be happy, and I also want to not overstock my tank- and the employee told me that 1 is fine by itself! Grr... So I bought 3. I went back today to get 2 more after reading y'all's feedback, and asked a different employee the same question. Got the same answer- 1 can do just fine by itself! Where do they find these people?! So anyways, I got my 2 more, and everyone seems to have adjusted quickly. They do still seem a bit shy, but not nearly as bad as before. I don't know if it's the increased group size, or if the "older" ones are giving the new guys confidence, but they seem much more comfortable. And, while I don't take everything they say at face value, aqadvisor.com says that a 10 gallon tank with a betta and 5 false juliis is only 81% stocked, and that's using adult sized fish. So I'm hoping that they'll be ok? I think (hope) that I have some happy cories at the moment, anyways! I can figure out how to upgrade later, if need be. I wish there were books out there on cories like there are on bettas! I'm hearing and reading so much conflicting information that it's hard to know what to believe.