An eviction notice served while someone is an owner of record of a house would not be enforced.
But, generally speaking, a divorce court retains jurisdiction to enforce its decrees after they are entered, and to clarify its existing orders.
Obviously I will need to refinance it and pay him his share of the
equity to get the deed solely in my name, which will take time.
Usually, the ex's duty to get the deed in your name would precede the duty to refinance it. A deed from one owner of real estate to another co-owner doesn't violate a due on sales clause of a mortgage or deed of trust, and the fact that both of you are on the mortgage or deed of trust does not mean that both of you need to own it. Unless the decree says otherwise, the ex would usually have to transfer the property even if the mortgage is not yet refinanced. But, not all decrees and not all separation agreements have the same requirements and provisions. The exact language in the governing document would matter.
If the decree doesn't say who gets to occupy the house, you could and probably should go to the court to have that issue clarified. Since violating a clear court order can give rise to contempt of court sanctions of incarceration or fines, often someone will comply with it without having to have the police intervene to remove someone.
Conceivably, if the ex did not leave, you could seek to hold the ex in contempt of court for violating the divorce decree if it implies that the ex's right to occupy the house has ended. If the ex is found by the court to have willfully violated a court order following notice to the ex, an arraignment and a hearing if the violation was not conceded at the arraignment, the ex could face incarceration and/or fines (contempt proceedings are rather complicated and exactly how you do this is beyond the scope of a simple answer).
You could also seek to hold the ex in contempt of court for not promptly signing the quitclaim deed (assuming that the decree does not condition the duty to do this on the refinancing of the house), but this would not be the best solution to that problem. This is because you could also go to the court and have it appoint the clerk of the court as the husband's agent to sign the quitclaim deed on behalf of the ex (assuming again that the decree does not condition the duty to do this on the refinancing of the house), if the ex failed to do so as required by the divorce decree by the deadline set forth in the decree, or a new court order, or within a reasonable time if not deadline is set. Then, after you were in title, you could bring an eviction action if necessary. It is much easier and faster to get an order directing the clerk to sign something on behalf of a party to a lawsuit than it is to hold someone in contempt of court.
You could also ask the court to issue a protection order prohibiting him from occupying the house at any time prior to the title being put into your name pursuant to the decree. This could take effect as soon as the protection order signed/authorized by the judge is served upon the ex. A protection order (in most states) is directly enforceable by the police, unlike most court orders which are only enforceable by bringing contempt of court charges in the court that issued the order. But, a court would usually be reluctant to issue a protection order unless there was a clear and present risk of physical harm, or severe emotional harm to you from a violation of the decree.
Also, while the ex wouldn't be committing the crime of trespassing by occupying the house, it wouldn't be unprecedented for you to persuade a police officer to remove the ex from the house after explaining the situation to prevent a breach of the peace between you, or because the ex's conduct in overstaying his right under the decree to be in the house constitutes "disorderly conduct" or "loitering" or "harassment" or some other minor offense in the opinion of the police officer, even if that charge might not hold up if prosecuted in court. But, there is really no way to compel a police officer to do that and it is close to the boundary of what a police officer does and doesn't have the authority to do.
Most of these remedies would take several weeks, at a minimum, to complete. A protection order could be done in a day or two, subject to a prompt follow up hearing within a week or two after the fact. A court might also rule on a clarification order on an expedited basis by setting a shorter than usual deadline for the ex to respond given the urgency of the matter.