Count me in. My 01 does the same thing but only at work. At first, I thought that I had hit the panic button on the fob, but after the 3rd and 4th, etc times, I was sure that wasn't the answer. It doesn't happen at home, and so far nowhere else. It only happens to me when I unlock it to go home from work. I, too, have the original batteries, but there is nothing wrong with them and the truck starts fine. I do see here, that this seems to be an '01 thing. Let's not let this thread die. I'm happier than you can imagine that I've found other people with this problem. If we can come to a conclusion, I'd love it.
BTW, I forgot to mention that my problem doesn't happen all the time. Maybe a few times a month, but like I said, only at work.