This does sound like a difficult situation.
One issue to keep in mind is that sole custody is only typically granted where there is a history of abuse, neglect, or some harm that will come to the child from awarding joint custody.
It might be difficult to gain sole custody in a case like this unless the husband was exposing the child to porn or sexual acts. If that was the case, then sole custody would almost certainly be appropriate with only supervised visitation
As far as solicitation, if it could be proven, the district attorney might be willing to prosecute. However, it would be a tough place for you to use e-mails which were potentially obtained unlawfully (ie. accessing someone elses account) because that could very likely be a crime in and of itself. If the DA did prosecute the ex for solicitation and he was incarcerated, that may increase the chance of getting sole custody.
If this information was helpful, please click ACCEPT
to give me credit for answering. Bonuses and positive feedback are appreciated if deserved. If you need any clarification, just post a reply. Thanks!