There are two things here:
First, your family situation. I am so sorry your mom is so ill and that everything has fallen upon your shoulders. I hope your mom will recover and that a responsible adult will step in to help you. I agree asking for a late withdrawal sounds like the best idea at this point, unless you think you can get a B or higher in the class or if you can attend the final without trouble.
Second issue: your expectations college is like high school, which it isn’t. Extra credit, do-overs, and fluff points are rare to non-existent in college. Expecting p-sets on Friday for a Sunday due date is extremely generous - most instructors at my flagship would either require the draft a week before or p-sets during office hours, generally the Wednesday or Thursday before a Monday due date, so that they have time to go over the exercises/draft, return it, and give the student 48h to correct/improve/edit. Everything you report indicates a mismatch between college professors’ expectations and yours and right now your instructor must be pulling her hair out about dual enrolled HS students.
BTW, do you start your emails with Dear Pr. X, I’m sorry to be bothering you but… And close them with Thank you. Sincerely,(name). Some freshmen don’t know how to write an email and end up sounding disrespectful without meaning to, compounding their problems.
The technical glitches are a problem for sure. Since you went through four different tutors, can you request they each provide a note testifying you did try to complete the assignment on time but were blocked by the glitches, for which you’re not responsible for. Perhaps IT can vouch for that too. The professor may be willing to consider your full assignment in that case especially since there must be quite a few students in your situation.
For Nov26, try to allocate a lot of time to the assignment in case there are similar glitches which is likely since everyone will try to log on and submit. If that seems impossible due to your family situation, see with the Dean of you can withdraw due to having to take care of your mother (a doctor’s certificate might help). This may be the easier way for you, unless you need this class to graduate.