OP, I don't know if you should just GTFO right now or if having a timeout (temporary separation) is of any benefit. That decision lies with you ultimately, and certainly what is best for your child will be a huge factor in whatever is to come ahead.
Trust is something that should be a given. I know, having been there before where it [seemingly] suddenly turns out that both sides aren't playing by the same rule-book. I say "seemingly" because frequently it is the case that the unassuming partner (the bloke normally?) doesn't realise things are so bad until a trigger event jolts him out of his stupor. Take your time, look at your life from the side like a passenger, remove yourself from your day-to-day for a short time if necessary. Consider the entire picture before making a choice.
Without Trust, there is no relationship. Things may be hard for a short time but ultimately you will be in a happier place - no point in her behaviour affecting you to the point where you are effectively having to spy on her to verify what she says. You are both better off without that behaviour - you more so than her.