Can't enter to win game because steam free weekend.

Don't participate in free weekends.

right click on Chivalry, click delete local content. Try to enter and if that doesn't work, refresh your steamgifts profile

Resync your account.

Now that the free weeekend is over you can resync, but you wouldn't have been able to do this until it ended. To avoid that situation the only thing to do is make sure you sync your account midweek, so that you're not stuck having to sync while a free weekend is on. Unfortunately, Steam doesn't provide a way for steamgifts to tell the which games are on your account and which are free weekend games. Aside from making a bot which syncs every weekend to use as a free-weekend-detector, there's nothing much SG can do. (A bot would be cool, though)

You've learned a lesson that has come up times before. Midweek syncing seems to be best.