The first thing is to assess what you mean by "a fully-funded DB plan". A fully-funded DB plan (measured using the funding relief rates of HATFA) may not have enough assets to cover the vested accrued benefits based on PBGC premium rates and owe a variable rate premium.
A plan that terminates under the PBGC's standard termination rules must have enough money (or know where to find enough money) to either buy annuities from a strong insurance company or pay lump sums. It will usually take a lot more money than the funding liabilities as measured for minimum funding purposes.
There is no longer such a thing as a "terminal funding vehicle". If the assets are not spent, as soon as practicable (which can still involve more than a year from start of the process to the end), for annuities and/or lump sum payments covering all plan benefits, the plan is not really terminated. After a plan has terminated, there are no unallocated assets, no ongoing group annuity contracts (other than a group annuity contract under which there are only purchased annuities for the individual participants).
After a plan has terminated, the onus for administering any annuities belongs entirely to the insurance company that sold them. The insurance company may (if the sponsor remains in existence) be able to obtain some help from the sponsor, but the overall responsibility belongs to the insurance company.
For more details, you may wish to discuss this with the plan's enrolled actuary (who should be quite familiar with the rules governing plan terminations). One presumes that as a self-described "DB illiterate", you are not the enrolled actuary for the plan!