Sounds like she's not over the cold. If she's able to roll over on her own you could try laying her on her stomach so that the mucous drains out. Or prop her on her side with firm rolled blankets, again, so the mucous drains out.
My grandchildren had difficulties with asthma and the other one with GERD. They slept best in the car seat set inside the crib. We first tried propping them up so that mucous, stomach acid would drain but they kept sliding down until they were flat again. They slept much better buckled in to the car seat.
If her colds over and she's just not able to handle the saliva I suggest that she will quickly learn what to do herself. It may be that the necessity to cough during the cold has become a habit and she'll learn that the saliva is different. Again, if it's saliva changing her position so that it will drain out her mouth instead of down her throat may help.
A red face can appear to be serious but as long as she's breathing it is not serious.
I'm not sure what you mean by choking. I think of choking in this sort of a situation as being unable to catch their breath for a second but that the increased coughing clears their airway. That is not a medical emergency nor do I believe it means a trip to the doctor.
If however your daughter is not able to clear her airway by increased coughing and turns blue and does this repeatedly over a short period of time then you should take her to the emergency room.
Feeling helpless gets to be a common feeling for many of us mothers. It doesn't go away once they become adults either. smile Be assured you are doing just fine for your daughter.