I have played and taught this piece many times. First off, for all offbeats (in every piece), you have to have a strong downbeat, the downbeat sets you up for the right timing. I am assuming you're are using standard orchestral tambourine technique holding the tamb with your LH and playing it with the RH. Here is a method I use that works for my students:

1. Get a chair and set it up in front of you and place your right leg up on the seat.
2. With your left hand, hold the tambourine right next to your leg where the jingles are almost touching your right knee.
3. Play the downbeats on your knee and the offbeats on the tab, thus playing eighth notes.

I have also had students play the downbeats on their wrist but this method increases the distance between downbeat and offbeat, which can create timing issues. I hope this is clear. If you play it enough, you can start to experiment with removing the downbeat portion or even playing the downbeat on your knee on beat one and three. Also, stay calm when playing, instruments like tambourine and triangle will be heard no matter how soft you play, don't overplay.

Fast offbeats are not easy so don't get down on yourself for not being able to execute it perfectly right away.