Your question does not mention whether or not your payment plan is ongoing or has been completed. If you have completed making all of the payments, then contact the other party's attorney, and they should be willing to provide you with a stipulation of dismissal with prejudice. If the payments are still ongoing, then the other party may be relucatnt to provide you with a stipulation of dismissial with prejudice. In the alternative, they may be willing to provide you with a letter explaing that the parties have resolved the matter and that you are currently making payments in a timely fashion. This may be sufficent for your potential new landord.
It is not possible to change the prior action's court marking. No doubt, your arrangement was made into a written "stipulation" (agreement) and signed by both parties; and the "without prejudice" was used so the landlord to whom you are to pay monies under the payment plan could go back to court if you later default under that plan. If you have paid off the entire monies owed under the payment plan, ask the attorney/LL for a written acknowledgement that you have fully satisfied the obligation and then use that document to show your new LL that there is no possibility that the action can be revived against you. Frankly, I don't think the problem is so much that no one wants to accept you because the action was "dismissed without prejudice"; it is more likely that the fact of the prior action having taken place raises the concern that you are a risk for not paying rent in the future. Good luck.
Unless I'm missing something, it would seem that your obligation to pay rent will terminate when you are constructively evicted by your complex "shutting down." I seriously doubt that they will sue you for your future non-payment of rent after they constructively evict you, but if they do, you would defend on the basis I've stated (i.e. that you were displaced by the fact that they "shut down"). Good luck!
If you think my response is the best response, it would help me if you would indicate that. Also, please note that my responses to question(s) are NOT legal advice from me to you because I am NOT your lawyer, you are NOT my client, and we do NOT currently have an attorney-client relationship. Thanks!
No. The case was filed and part of the court records. You were evicted due to nonpayment of rent. The matter was settled. This does not take away the fact that you didn't pay rent. You can explain the situation to prospective landlords but it doesn't go away.
You did not mention whether you have paid up on your payment plan. If so, a satisfaction of judgment can be filed. That would act as a dismissal with prejudice, and show you paid your judgment. If another complex turns you down, because of non-payment or an eviction, that is their decision.