Something that occured to me after my earlier post. You could accept applications during the month, just make it clear that players are not eligible to compete during the month they are accepted. That wouldn't be too difficult as the date each player joined the league is right next to their name.
Other ways: some players put emails or other contact information in their profiles. Or a post in the forum asking so-n-so to contact you. A contact email for you in the league info so that players can ask you privately (use a throwaway freebie account, there are any number of sites that offer them).
As for how people react to not hearing from you, that's human nature. Humans are social animals and if we don't get a reaction from someone we tend to assume the worst and our imaginations have no checks. I'm not trying to upset or antagonize you here, just trying to help.