My spouse and I do not live together. The children reside with him. We are not legally separated or divorced. I am paying him child support. Can he just "drop the children off" and say they are in my care and then take off for a few days even though I told him I was not able to watch the kids during that time? He has done this on more than one occasion. I tell him to get a babysitter.
Until you get divorced, you and your husband, no matter how separated you may be, are both responsible for caring for the children or seeking help in finding other alternatives. If you want more certainty about who has what obligations, you'll need to bite the bullet and get a divorce decree that includes a separate parenting agreement spelling out who does what, as well as what happens if somebody falls down on the job. When you do file for divorce, you'll also be able to ask the court for a quick decision on custody and visitation. For more information, see Temporary Orders in Family Court: How to Get Quick Decisions on Support and Custody.
In the meantime, or if you really don't want to file for divorce, you can negotiate with each other to create a written parenting agreement on your own or with the help of a mediator. Your agreement can set out fixed times for visitation if you wish. For more information on creating an agreement, see Building A Parenting Agreement That Works, by Mimi E. Lyster (Nolo). For more about mediation, which might also be a process that could help you come to an agreement, see Divorce Without Court: A Guide to Mediation & Collaborative Divorce, by Katherine Stoner (Nolo).