...I would get that board out from underneath there. You can't get a full picture with that there. "Falling apart" could be an omen to what is under it, or it could just be in bad shape. I would see if you can get under there and remove the board before someone comes in to take a look/do the work. They could find stuff you didn't take into account and have your steps partially disassembled. Then you'd be in a little of a rut.
Is the damage all the way through? Has someone told you that they are unsafe? The reason I ask is that you may be able to buy a little time here if the stairs are still good. You certainly want to repair them, but is it realistic to get someone in this week before the weekend? If they are structurally sound for moving in, then I'd move in and make them at the top of your list to get fixed...after you move in. You can get more than one quote. While it might seem like a small job, you'd be surprised at the range of prices you may be quoted.