[Fiddlebog]

sounds to me like the problem is the steering lock, since the steering lock does not re-engage until after you remove your key from the ignition. I highly doubt it has anything to do with the ignition switch.
These are complex systems with more than one component. You can try replacing the ignition switch, especially since those are so cheap and very easy to replace, but the real solution will likely be cleaning out the steering lock (some graphite also wouldn't hurt).