What can I do to prevent false abuse allegations from being made against me?
The only certain way to prevent false allegations from being made against you is to have witnesses present during any potential time you would be vulnerable to being accused. This would be:
Needless to say, ensuring that you have round-the-clock witnesses is hard to do. If possible, enlist a trusted friend or two who understand the gravity of the situation and are willing to help you out. If you cannot get someone to accompany you, you may be able to use a camcorder to help document events.
If you suspect you're likely to be falsely accused of improper behavior with your children and cannot get someone to accompany you, take your children to some reasonably public location or go visit with friends.
The main thing is to be able to show that you weren't alone with your children for any length of time. If the threat of false allegations is very high, you may want to consider not exercising your parenting time until you can arrange for a witness to be present.