You mention you think she is getting rid of evidence on her phone.
Do you have any evidence of adultery? Anything that is court admissible, or even anything that is not?
Also, did you check with the lawyers you met about the implications of adultery on divorce in your area / with local judges. In some jurisdictions it can void spousal support, and also give you a lot of ammo to force the cheating spouse out of the house, often without the kids.
In most areas though adultery officially has no impact on divorce.
There are a few approaches you can consider. 1. See if the adultery can reduce/eliminate your support payments. Check if you are in a fault or no fault jurisdiction. Also, ask around with different lawyers and see what they think. Even in no-fault jurisdictions, you can often find judges who would beg to differ. It would be good if you find a lawyer who thinks that whatever you have or can get as evidence of the adultery would help your case with alimony or custody if this goes to trial.
2. Keep evidence as leverage. When you get her absurd papers, tell her that your lawyer wants to counter sue based on adultery grounds and make this public record. This can be strong inducement to settle in mediation. However, given how you describe her state of mind she probably cant envision herself in the wrong, or losing.
Also, you have not responded about finances. Have you separated them yet? I assume you are not paying for her new phone? What about her L? Anything she buys or commits to now is 50% your expense until you get things formally separated, depending on the definition in your area.
Also, remember that she is obviously not thinking about a future where she has to work full time given her shortsightedness. When she hits you over your work schedule on custody - well she has no idea what her work schedule will be, but she will definitely have one! It will be up to each of you to work around that yourselves.
However, if you have right of first refusal this might give your W an in to keep your D from end of school until you can wrestle her back each evening. I recommend getting RFR about 99% of the time, but you (or rather your WW) might be part of the 1% where I can see this becoming an issue.