Well, if you get under there and see alot of rot, all this info I'm about to type will be irrelevant...I'm assuming that the joists are in decent shape, just sagging from the weight of time. I'm also assuming that you have enough access under there to get full length joists inside, and that the support at the ends is solid.
If you sister them properly, and have them supported by at least 2 inches on each end, they won't need more support in the middle. First, you go down there with extra lumber to place under the area across the sagging joists (doubled up), and some to place under the jack. You have to make sure it will even move without messing anything up.
If the joists will move up to where you want them without much problem, you take the jack out of the way, make the cuts, and tack them into place with 1 or 2 nails, or lay them on their sides in the joist bays. You'll never get them in with the jack in the way.
Everything goes well and you should be able to jack it up, install the joists and nail them in a good pattern...even glue them if possible. When you're finally done and remove the jack, it shouldn't drop much.
It's one way of doing it, anyway....
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