Insurance booked through
Disney Cruise Line can be added up until the final payment date. In fact, all insurance booked through
DCL is not actually purchased until the final payment date, since there are no non-refundable expense before that date (except for concierge/special bookings).
For insurance purchased on your own, you would need to check with the company, but most insurance can be purchased or upgraded until close to the sailing date. However, insurance typically does not cover known issues, so insurance purchased now might not cover disruptions related to COVID-19. If you already have insurance and then upgrade your room (meaning the cruise is more expensive), upgrading the insurance coverage to match your revised costs may result in an additional fee.
Having paid in full will not affect insurance rates or ability to purchase as long as you could still get a full refund (that is, before the required final payment date).