People who have nothing to hide, hide nothing. A remorseful WS would WANT you to look at their phone or any other part of their life...forever, if that's what it took to earn your trust and make you feel safe!
Sorry, but she is not remorseful and is still hiding something.
she feels I need to earn her trust back
Hurt, this ends when you say it ends. It seems very probable that she is still in contact with the OM. Have you told his spouse? This is paramount; she deserves to know what is going on within her own marriage and she can help shut this affair down. Do you have a home phone where the boys can get in touch with their Mother if necessary? Then shut down her damn cell phone. She'll have a hard time continuing the affair on a land line.
I would also see an attorney pronto; file that divorce. She will continue to treat you like a paycheck as long as you allow her to. I know this hurts, but you don't need to have the added hurt of her treating you so poorly.
Seems like you may have to have another discussion on what it really means to each of you to "try to work on the marriage" and how you can get there (one of the conditions being total transparency).
File and serve her with papers. She knows what she's done. If there is any hope of her dislodging her head from her arse it is with a dose of reality & consequences. I would not hold out much hope for that though.