Hi, sorry to hear you're going through this.
First off, I wouldn't say running is the best way to deal with it, it's just escaping. However, getting space could be good for you. Can he take care of the children so you can go away, telling your children you're visiting someone?
The ways he's lied to you, cheated on you and his drinking problems don't paint a good picture of him. He sounds very difficult and inconsiderate.
You say your marriage has other issues. Can you talk about how your marriage is generally? Is it really great and worth being in apart from the cheating? If so, maybe you can deal with the infidelity (it isn't easy) and try to move forward. If it's not so great otherwise, this might be more difficult to recover from. Staying with someone just because you have children together isn't a good idea. In my opinion, the fractured household hurts children more than a happy single-parent household.