I think I overloaded a wall outlet and the breaker never tripped
This is not good! What size breakers are on those circuits?T here are definitely signs of arcing and burning on some of the neutrals that could be caused by loose connections. Turn off the power to those circuits, cut the burnt part of those white wires off, strip off 1/2" of the insulation and insert the wires in different holes in the neutral buss and torque connections to specified values. Check to make sure that if there are any multi wired branch circuits the two hot legs are on different poles. If they  are on the same poles you will be overloading and burning the neutrals. Is this something you just happened to find? Maybe it's an old problem someone previously did a poor job of fixing. Or were you experiencing problems and smelling burning and went looking for the problem?