I would get a few more quotes, also do not cut the existing joists. You will want to marry up another joist next to the existing one, glueing and screwing the two together. Run the joist from girder to sill plate. Is your house vinyl siding? If so I would remove the bottom few courses of siding, cut the plywood and remove then remove the sill plate from the outside and replace. You will want to temporarily brace the floor joist from the crawl space. Once the sill plate is replaced then you can slide the floor joist in from the outside up next to the existing floor joist. Only use treated for your sill plate and then regular yellow pine for the floor joists. If you use treated for the floor joist you will run into other problems down the road. Then as long as you have your house treated for termites you should be ok...if you have a water issue you need to eliminate that either with gutters on the roof or a French drain, just depending where it's coming from.
