How about something entirely different, more along the lines of free agency? This should just be regarded as a general idea, with lots of possibilities for specifics.
At the end of the season, a team can sign up for a scouting combine. A scouting combine would consist of 10 teams (not necessarily from the same league) and 20 prospects (4 at each positions). The 20 prospects would be divided into 4 squads of 5 who would play a half in each of 3 games. So in Game 1: red+white v. blue+gold; Game 2: red+blue v. gold+white; Game 3: red+gold v. white+blue.
Games would be available for replay, etc.
After each game, each GM can request to have one skill revealed for each of 5 players. All GMs will be able to see this information.
After the 3rd game, GMs can rank from 0 to 5 players. Players who are ranked #1 by at least one team are selected, with ties broken either by lot or some ranking procedure. Once a team has a player their other picks are disregarded. Teams that miss out on their top ranked players skip to the 2nd pick. This process is repeated for the remaining picks.
Players that are passed over go into a 2nd chance pool. Teams can sign up for a 2nd chance scouting combine, that works just the same as the first combine, except it will be made up of players passed over in the first round combine. Players passed over this time, will go to a 3rd chance pool. If they wash out there, it is the end of their career.
The combines could happen at any time, so late comers such as myself won't be disadvantages. A team can sign up for a scouting combine when they are ready to participate, and it is a process than can be completed in a week. Teams have equal information, since they will be able to have seen all the players perform, plus have limited information about their skills. By mixing teams from different leagues, you mitigate the possible imbalance of a draft. where there might only be a few teams actively drafting in one league, and 16 teams in another.