You can only submit a new sorting app at the platform if you have currently been away from HiH for at least two months. So if you left the community in early June or prior to that, without having been a member in the meantime, you can submit a new app.
Since you are listed as a member of your common room and the main comm, that doesn't seem to be the case, so that leaves the option of an appeal.

Knowing what other house you would be suited for is not necessarily a requirement, we can work around that. Though for obvious reasons it often makes the process a little more straightforward if you can give the judges some sort of idea.

Let me know if you have any more questions. :)