Its always best to power down a device before working on it, so I wouldn't suggest trying to replace the digitizer with the device still on to prevent any further damage.
It does sounds like the digitizer got damaged, worse case scenario the FPC connectors on the logic board where the digitizer cable plugs into got damaged. But you won't know till you try the new digitizer. There is 1 single screw holding the logic board down on the battery contacts. Simply remove this screw and with a plastic thin pry tool gently lift the logic board up slightly so its not touching the battery contacts anymore (there is a little adhesive holding the logic board to the frame so it will require a little force but not too much), this will power down the device. Ive attached a pic below showing the single screws that needs to be removed. Ignore the 2 screws on the right of the pic.
Its always best to power down a device before working on it, so I wouldn't suggest trying to replace the digitizer with the device still on to prevent any further damage.
It does sounds like the digitizer got damaged, worse case scenario the FPC connectors on the logic board where the digitizer cable plugs into got damaged. But you won't know till you try the new digitizer. There is 1 single screw holding the logic board down on the battery contacts. Simply remove this screw and with a plastic thin pry tool gently lift the logic board up slightly so its not touching the battery contacts anymore (there is a little adhesive holding the logic board to the frame so it will require a little force but not too much), this will power down the device. Ive attached a pic below showing the single screws that needs to be removed. Ignore the 2 screws on the right of the pic.
https://d3nevzfk7ii3be.cloudfront.net/igi/AvOTpM4Hg2LSFZQM.huge