You have two options for using only a single deck for Alpha/Omega cards.
Option One: Have a single deck for Alpha & a single deck for Omega. Every player uses the same decks. This is easy and cheap, plus it actually simplifies play, but has the drawback that you're going to get a lot of Alpha powers that aren't really thematically appropriate for your character.
This is also the official solution to this "problem"; from pg66 (emphasis mine):
During play, sometimes you'll draw from your own player deck and sometimes you'll draw from the Game Master's deck. If you haven't constructed your own deck, all your draws come from the Game Master's deck.
There's no reason you can't do the same with the Omega deck.
Option Two: Let each player build their own small Alpha & Omega decks as the game describes using the available supply of cards (presumably the DM's). While this doesn't require any changes to the rules, and allows characters to keep their own personal pool of appropriate Alpha powers and preferred Omega items, it does mean that each card can only be used by one player (since AFAIK there are no duplicates in the decks that come with the base set). If this results in conflict over cards, you may want to use a simple drafting system for players to build their Alpha decks.