Umbilical cord question

My baby is one month 3 days old, and his belly button stump fell off before time. He accidentally pulled on it when I was changing his diaper. Took him to the er and they said it happens more than one might think. It's now been about 2 & a half weeks since then. The thing is, it still smells really bad. I could smell it through his clothes. His doctor saw it at his two week checkup and said that it was normal for the stump to fall before time. I gave him a tub bath two or three days after it fell off because it looked healed. Then when I was changing him, I cleaned it with a qtip and when he stretched, I noticed that the belly button goes deeper than what it seems at first glance. There's skin showing which looks to be healed and dry, but it goes deeper. There's a space underneath that, if it makes any sense. You can't see it when he's still, but you could see it when he stretches. I clean in there too a bit. I don't put the qtip in too deep for fear of hurting him, but that space looks like it's not fully healed yet. It's always a bit wet. I can't take a picture cause he's asleep, but I clean it every time I change his diaper. I fold his diaper so it won't touch his belly button and so pee doesn't go in it. Even after that, it has clear discharge sometimes and it smells really bad. I know it's not normal, but should I be too, too overly concerned about it? He doesn't have fever and doesn't seem to be in pain. I check by putting a bit of pressure around the area when I clean it and he doesn't move as if it hurts or anything that indicates it's painful. Not even when I touch his stomach or anything. I don't know why there would be discharge though. Has this happened to anyone else's baby whose belly button stump fell off before it was supposed to? Did it happen because it fell off before time? Does it take longer to fully, fully heal in there because of it? Is there a better way to clean it? I feel like I'm doing everything wrong because I've never seen this happen before. The doctor didn't say anything about it that time. He did pull on the outside of it to check it well, but didn't say anything.