I have a 1900's house with sagging floors and rotting support beams. Some sagging is normal for an old house, and unless it is drastic, I would suggest just letting it sag a little. As for the rotting beams, they should be replaced from below. You can do this by removing old beams and replacing them one at a time, if the structure an support the load for a short time, with the one beam missing. Otherwise, you will need to temporarily brace the structure from a position where the new beam can be put in.
Most structures can stand to have a floor joist missing for a day.
Just make sure you don't have a bathtub full of water directly above. My home was built in 1927 and I have some floor joist that have split and twisted. The original joist has a small cut-out and they sit on top of the foundation. If you are placing new joist against the side of old I like to use double joist hangers at the rim joist. The 1st floor has a laundry room and lavatory.