You did not indicate if the person you spoke with was just a staff person at a desk or someone with formal administrative responsibility, i.e. an assistant principal or deputy superintendent. You also apparently were doing a lot of things on-line which can be very, very problematic.
I recommend that you contact the school district's superindentent's office and arrange a meeting with someone with the authority to review the school assignment issues for your son. There may be a community organization that works on education issues that may be able to assist you as well. Some districts have rules around mid-year attendance changes but most should be able to accommodate a child moving from one district to another. It is also not clear if the school closest to your home is a charter or some other type of specialized program that may have admissions rules that may not follow the rules in your old district which probably meant that children were assigned to the school closest to their home. Knowing the special rules is something that is not always easily accomplished with on-line registration. Some things particularly enrolling a child in school should be done in-person.
This is not a substitute for a consultation with an attorney with a background in education law in your new community.
It appears that you were misled by this staff person. You should talk to the highest level you can meet with in the District office. Perhaps that will get you into the school, if not right away, when a spot opens up again. An attorney can assist you.