This might work, but you need something for them to fight over instead of championships.
The alternative that comes to mind is to do periodic tournaments with a trophy/medal and cash prize. The "regular season" matches would provide standings that determine who qualifies for each tournament and how they're ranked. There could be different styles of tournaments throughout the year.
Money In The Bank: winner can cash in the briefcase to enter one tournament during the next year that they didn't otherwise qualify for
King Of The Ring: straightforward single elimination tournament
Battle Bowl: random tag teams for the qualifying round; winners in a battle royal
Crockett Cup: regular tag team tournament (maybe you don't a regular tag team divsion so everyone has to find partner for this tournament each year)
World Cup: wrestlers representing different countries
US Cup: wrestlers representing different states
Extreme Rules, Rage in a Cage, etc: themed tournaments where all matches are of the same type
Royal Rumble: winner gets a spot in the Wrestlemania tournment
Wrestlemania: 8-man single elimination tournament featuring winners of the 6 most prestigious tournaments, the Royal Rumble winner, and the wrestler with the the best record for the year
You could still have feuds and non-tournament side matches.