Can I use a one time pad key twice with random plaintext?
Yes, encrypting two different random "plain texts" with the same "pad" is indistinguishable from using two different random one time pads for encrypting the same plain text. You get perfect secrecy in the latter case, so you will get corresponding secrecy in the former case as well.

However, usually there is a functional difference between the key and the plain text that the above argument doesn't account for. More precisely, the recipient will usually "do things" with the plain text once decrypted, and in that respect the above argument might fail. 

For instance, if the random messages are symmetric encryption keys for a cipher that is vulnerable to a related key attack, it is obviously not safe to transport those messages encrypted with a reused pad.