Clue - getting suggestion right early on, what happens?
An important strategy in Clue (and its variants) is to frequently use cards from your own hand as part of your Rumor. Doing this can give you more information, and it also means that the game doesn't degenerate in the way you describe: when no one is able to disprove your Rumor, they won't all race to the center (unless they're lucky guessers, or you don't have a good poker face) because they can't know whether the cards are in the center envelope or are in your hand.

However, original Clue (assuming these are the original rules) does also solve your problem by allowing you to make an Accusation on your turn from anywhere, including immediately after your Rumor turns up no disproof.