Last winter when I used the heater I sometimes.

Last winter when I used the heater I sometimes noticed an antifreeze smell. This winter apparently the heater core sprung a big leak because when I first noticed it, half the floor was wet but I didn't realize at the time why. Well now its time for the car to go to the shop for appointment made 3 wks ago and I'm checking my fluids under the hood and find out the radiator is a qt low. I add the water, start the car to warm it up and run the heat on high to warm the cab for the drive. After a bit I shut it off and then when ready to leave, by mere chance, I noticed the passenger floor and mat are litterally standing in water. Now for my question, I have no way to tow my car, I know that the heater core can be bypassed but I dont really know the first thing about reaching and removing those hoses and with a bad back I dont want to attempt it.--So if I refill my radiator, and put the heater control on off with the temp all the way over to the cold side and fan off. Can I drive the car to the shop or will I still lose water thru the bad core?
Thank You Much as this is stressing me out.