Will using a flood guard on cellar drain cause excess storm water to go to upper floors?
If that's your plumbing layout, presumably it would overflow at the basement laundry sink if it was going to overflow with the floor drains successfully valved.
Whether it would will depend how much pressure there is - while your flood was only a couple of inches, it was spread out from two floor drains across however much basement area it flooded, so there might well have been more than 2 inches of back pressure if contained in a pipe rather than allowed to spread out.
Assuming your drains connect to a municipal sewer, requesting the agency responsible for maintaining that sewer to correct the problem of it backing up during storms (perhaps indicating a partial blockage/collapse or need to be cleaned, or infiltration [storm water getting into the sanitary sewer] if it's not supposed to be a combined sewer/storm sewer) would be the best solution.