@Ray A Delfi
@Ola Dantis
@Jay Hinrichs
Ray it sounds like you're already thinking of getting a structural engineer out there. At least here, there aren't many that practice actively in residential work, but they do exist.
From that perspective, my comments can be just taken 'for what they're worth' or perhaps to give some perspective as you get an engineer. I do think this is a project that justifies an engineer.
As Ola touched on, or mentioned, you will want to determine if you have any settling. That could be a major driver for whether you'll need to do much / anything with the existing masonry wall. If it's settling, the common fix there is to shore up the footer system, but that entails quite a bit more work as compared to simply shoring up your joist system.
So for some background, this type of setup is how buildings 'used' to be built, in that back in the 1800s and early 1900s masonry walls (as in, structural walls) were the common way they were built when you saw masonry. It still on very rare occasions is done, and also on rare occasions for retrofit/modification work, you may still look for opportunities to support your structure/joist systems etc with the masonry - however sometime in the early 20th century pure masonry started going out of favor, for structures being built entirely with their wood/steel, and the masonry you saw was then basically an aesthetic detail that was 'tied' to the structure holding everything up. The mechanism to do that is referred to as a wall tie.
Our house was built in 1929 in Des Moines IA and had already transitioned to this setup. Luckily that was the case as we had a very significant structural (outward bowing) on the house's north side, due to water entrance in the cavity behind the bricks, causing the wall ties (which are metal) to rust out and fail. Had these bricks been structural in nature, like your situation, that bowing may have been a fatal flaw and caused us to have to knock the house down.
@Parker Eberhard
I think Parker has some good thoughts there; I'd like to see more pictures/details of his recommended approach. I suspect there may be a good design there but you may think about getting a hybrid design from some of the best ideas.
I think Jay mentioned something that may be your most economical solution given that you're going to be replacing some joists, anyways - in a nutshell, you'd be removing joists that are bad (potentially all) - cutting existing joists you're keeping out from the masonry wall (no longer touching), and building yourself a solid ledger type system that you can use hangers for your joists.
I don't know the geometry of the room, but if you can picture a square or rectangular shaped room, you could get by with 4 new footers. Digging those footers will be some work, for sure. Harder work if you're not removing joists, from an access perspective.
I also was not formally trained as a structural engineer, so my terminology may not 100% be 'on point' but if you can imagine a new 'ledger board' system that is supported by these 4 footers, and then using joist hangers to support the new joists, that hopefully gives you the visual.
I personally would not mess with supporting something running laterally underneath existing joists, given that you are replacing some joists.
Yeah this type of project kind of separates the real deal investors and contractors in my opinion from the folks who aren't going to get there. Thanks for posting and good luck with the project!