Divorce is a very difficult event to cope with for everyone involved, even if it was the best option for what you were going through as a couple. Children can act out when this type of change occurs.
The most important thing to do is to make sure that your children know that the divorce was not about them, that you both love them, and that you are there for them no matter what. If you have already said it, say it again and again. They will need consistent reassurance. Even when you are frustrated with their behaviors, tell and show them that you love them. Address the lying behavior by issuing an appropriate consequence such as an early bedtime or adding a chore. Also, talk to your child about the importance of honesty. It might help to start off with letting your 8-year-old know that you understand that this has been so hard for him. Ask if there is anything he wants to talk about and let him ask questions. Be honest with him, as much as is appropriate.
Make time for fun things with your children. Family time may not sound appealing right now, but it will help you and your children tremendously to spend quality time together. Consider family counseling to help you get through this adjustment period. Due to the fact that parents are so emotionally drained from a divorce, it can be hard for them to put the pieces back together for their children. Sometimes getting an outside perspective is very helpful.