If you can, could you have a shower and massage the area as much as possible starting from under your armpit downwards and let the breast drain some. Afterwards wake Bub for a dream feed off that side to further the massage work you've done. Hang in there, it's terrible. If you feel worse tomorrow go and see someone so it doesn't get even worse.
ETA: hot only ever worked for me too not cold but put as much pressure as you can withstand.