It's all how you present it, actually...
Title 16, Chapter 12, Section 21 (16-12-21)
(a) A person commits the offense of gambling when he:
(1) Makes a bet upon the partial or final result of any game or contest or upon the performance of any participant in such game or contest;
(2) Makes a bet upon the result of any political nomination, appointment, or election or upon the degree of success of any nominee, appointee, or candidate; or
(3) Plays and bets for money or other thing of value at any game played with cards, dice, or balls.
Subsection number 3 is the one that would potentially apply to you. Notice that there is no wording which declares that gambling only occurs when the house takes a cut. Gambling can occur when people are playing and betting for money, regardless of the existence of a "house" or whether it takes a cut. The definition of "bet" is given in an earlier section:
Title 16, Chapter 12, Section 20 (16-12-20)
As used in this part, the term:
(1) "Bet" means an agreement that, dependent upon chance even though accompanied by some skill, one stands to win or lose something of value. A bet does not include:
(A) Contracts of indemnity or guaranty or life, health, property, or accident insurance; or
(B) An offer of a prize, award, or compensation to the actual contestants in any bona fide contest for the determination of skill, speed, strength, or endurance or to the owners of animals, vehicles, watercraft, or aircraft entered in such contest.
All you would have to do is spin this as a Texas Hold 'Em contest. People put in money and buy chips. Make those chips different values than the money they put in. Then have 1st, 2nd and 3rd place prizes; don't have the winners split the pot in any sort of manner. Just have set prize amounts based upon the number of participants. This way, it's no different than people paying entry fees into a contest/competition. They stand to win a prize at the end if they are more skilled, but they're not placing bets and increasing their potential winnings in any sort of gambling sense.
Additionally, I assume the organization is a 501(c)(3), as Section 503(c) deals with the FDA. Regardless, non-profit organizations are held to the same standards when it comes to gambling, so that doesn't have any affect on the situation.