I ran into this once, too. In my LED lights, there is a potted blob of plastic, one per section, that doesn’t open. I used a dremel tool to get into it (ground away enough of the plastic, and found a high-wattage resistor in there, and maybe a backwards diode? I don’t remember. I replaced the resistor and potted the whole thing up again with Sugru, and it worked. Sorry I didn’t record the value of the resistor or take pictures.
The reason for the two different size sockets is that if you look at your strand, every so many lights three wire come into the socket instead of two. I think (and this is only a hypothesis) than the shorter sockets are to allow room for the connection of the third wire. Why they didn't just use short sockets for everything I don't know. But it seems likely to me that the short one would work everywhere. I have not tried it.