If we need to move out early, how do we get our landlord to re-list for rent?

1) You can't force him to relist at any particular moment or in any particular time frame. His duty is only to make "reasonable" efforts to re-rent the unit. If you break the lease improperly and early and he sues you for the remaining rent due under the lease, if you don't believe that he has tried to "mitigate" (or reduce) his "damages" (or losses) by making reasonable efforts to re-rent, you could raise that as a defense in court. A court can reduce the amount of rent he could get from you if it agrees he did not make reasonable efforts, though since, even with reasonable efforts, it takes time to re-rent, assume he'd still get at least a few months rent from you.
2) If the mold conditions and pest infestation is sufficiently bad so as to render the unit uninhabitable (not just undesirable; uninhabitable) by the average reasonable person, you may be "constructively evicted," or effectively forced out by the landlord's violation of the "implied warranty of habitability," or obligation to provide premises fit for their intended purpose (residence) and habitable. But it is a high bar to be constructively evicted: the courts often expect tenants to put up with quite a lot before concluding that they can't live there. You are strongly advised to speak about the situation in detail with a local landlord-tenant attorney; bring with you extensive photographs of the issues, maybe video, any exterminator or test results you have, etc. so the attorney can help you understand if you'd be entitled to leave--or alternately, the lawyer may be able to get the landlord to fix the conditions.