Yes, you can run the pump until no more water comes out. What water is left in the tank shouldn't hurt anything - you can put some RV antifreeze in there if you aren't sure.
Make sure the water pump is empty when you are done! or... also run RV antifreeze through it.
Meanwhile, figure out what is wrong with your tank drain valve. My freshwater tank drain valve actually needs to turn 180 degrees to open, not just 90 degrees, so make sure you are turning it as far as needed. It may not drain fast, so give it a chance to see if it is draining when you turn it.
Otherwise you may have to replace it. Not that big a job - assuming you can get to it!