"deep cleaning" is never an urgent issue. You can wait several days or even months until you get the insurance. Or, find find a subsidized clinic or a community college dental hygiene program, where they do it for cheap.

"if you don't do on time you might lose a lot of teeth" is what a private or corporate dental practice would say to make immediate money. If you go to a non-profit or to a good dentist, they will not scare you. Nothing serious will happen within days as it is a slow process to develop a situation needing deep cleaning.

No worries. I m not a dentist but I interact with many regularly.