Is it ok to asking several profs at the same time for a letter of reference on short notice?
Because of the workload professors are under, it is not an uncommon problem. The only way to address the problem is to contact multiple professors for a recommendation; I know other people who have contact multiple professors for this purpose.

You have only asked the Professor if they would be willing to write a letter of recommendation. This means that they have not already invested any time. So, If you get multiple recommendations, you can simply email the other professor(s), "Thank you very much for agreeing. Another professor has already offered to write me a recommendation letter, so to save yourself some time you don't need to write one for me too".