OP, frame your opponents biceps to keep him from getting head and arm. Also, if you can immediately get onto your side, you can put your bottom side arm's hand on your forehead... kind of like a forehead slap (doh!). This makes his crossface slide right over your hand and over your head. Someone else already said that you should be active. That's very important. If you're working to get the lockdown and you pass up an open sweep, that is silly. Start grabbing for the toes even if your lockdown isn't in... get the lockdown in while you are working sweeps.
If they knee slide straight away, swim your left hand in and get double underhooks. You can go straight to dogfight from her if your double unders are perfect.
If you get crossfaced after all that, you have to stomp immediately. Stomp like your life depends on it and push up to butterfly quickly. From there you are on 100% offense with that overhook.
Also, re-read your books and see what they say about halfguard options/transitions. Lockdown is the treetrunk of the system