I seem to have found a workaround to the problem, now that I have narrowed down the problem.
The issue occurred whenever the network adapter I wanted to use had been turned off prior to me wanting to use it (either by using the switch on the laptop for the wireless, or letting Windows turn either device off to save power). I already told Windows not to turn the devices off. But now if I want to use the wired connection I tell Windows to disconnect from the wireless network instead of just turning the wireless card off. I haven't been able to test this too extensively yet, but it seems to be a stable way to switch without requiring a restart. Later I will see if I can get online if I tell Windows to disconnect, turn the wireless...