Sounds like you have a bad case of sag. Don't feel bad, sagging is just part of old age as many of us can attest. Your concern about getting the floors level and stronger without causing damage from sudden shifts is valid. Another way to approach this might be to use adjustable floor jacks and a beam that cuts the current joist span in half. The idea is to install a beam supported on adjustable columns, perpendicular to the joists. This beam as it is very slowly raised into place takes up the load at the center of the joist, effectively making it half as long.
This approach actually uses less lumber, but requires a larger beam and adjustable columns with steel plates or poured reinforced footers to hold the loads. The idea is to initially just bring the beam into contact with the floor and making very small adjustments over a long period of time until the floors become level.
After you achive a more level floor, additional lumber can be sistered into the joists if necessary. You mentioned the joists are in poor condition, but did not specify if it was rot or just age-induced deflection.