No, its not normal, sounds like some kind of bacterial infection, if there's oozing gungy stuff in the stem, unless you're just seeing the usual wetness or sap appearing from a wound. You haven't described whatever the bugs were on the leaves, but if the plant is still alive and growing, and not worsening day by day, it might be worth spraying with something like Roseclear 3 at the fungus treatment strength (take outside to spray and bring in again immediately). The other thing to check is the compost - turn it out, check for bugs in there, replace with new compost, having first washed the pot thoroughly if you don't use a bigger one.
I'm not sure either how one of the stems "broke off", they don't usually do this unassisted, so that might be a sign of some serious infection. These plants are usually low maintenance, trouble free and easy care, so this is unfortunate, but they are at least easy to replace (proper name is Dracaena marginata)