You are trying to solve the wrong problem.
The problem is, this year we do not have enough personnel to grade the homework.
Note what I've emphasized. The problem is not "how to make do with not enough personnel", the problem is "not enough personnel". If you try to somehow hide that fact with some kind of a "hack" solution - which you yourself don't think very highly of - you'll just make it even harder to solve the problem.
(While not the focus of my answer, I'll also say that your self-grading system is unfair to students. Feedback from a trained academic is of critical importance to their education, especially the feedback about what they have not thought of themselves, or regarding alternative possible solutions which they may have stumbled upon, or errors they make trying to establish what's correct in itself, and so on.)
"But", I can hear you saying "The dearth of homework graders is a systemic, non-technical problem - not something I can solve merely by my sharpened intellect. And it's not as though I can just tell my department 'we need more people', since other courses probably have a similar issue too."
Well, to that I would say:
It's time to unionize.
(or to shake up your do-nothing shell of a union you may officially have.)
Faculty or University management has allowed itself to cut back on the number of positions beyond the necessary minimum. That has been possible for some combination of the following reasons:
- It can make decisions without consulting with the faculty
- It can force its own decisions on the faculty
- The faculty do not have enough sense of professionalism / perspective on the pedagogic process to voice objections when consulted
- Faculty representatives are co-opted by management
- Faculty representatives are not subjected to oversight / do not really need to answer back to their public
These are all issues that a faculty union is supposed to - and can - address. Note that I didn't say you need to start the socialist revolution and take over the world, but if you had a working union, you could (in order of the above issues):
- Demand from, and possibly force, management to only make such staffing decisions when the faculty agrees that they're reasonable
- Allow for individual course professors to not have to accept the allocation of TAs/homework graders/lecturers they were given from above, i.e. for the individual (like you) to say "that's not enough" and for there to be a binding procedure for addressing this beforehand, rather than you being stuck with the current situation
- Educate each other, especially new faculty, and more especially new graduate students who become junior faculty, about things like obligation vis-a-vis the students and vis-a-vis your colleagues, what the faculty believes should be the reasonable workload and why, etc.
- Make sure that faculty representatives to decision-making processes are at least somewhat beholden to the actual faculty and its interests, e.g. through elections, and also:
- Make your representatives report regularly to their colleagues, both in writing and in the occasional meeting, and that you all make certain decisions together rather than leaving everything to their on-the-spot personal judgement.
If the above happens, I'm sure you will have both the will and the means to rectify the lack of sufficient personnel. In fact, collective power could allow you to do things that are personally less possible, e.g. get the faculty to invest in automatized question generation and grading, at least in some courses (not algorithms probably), which could free up some of the teaching staff.
PS - Since you (seem to be) Israeli, I recommend you read a few relevant chapters of Dr. Daniel Mishori and Dr. Anat Maor's anthology about Exploitative Employment in Israel. It's available on-line for free (Hebrew only though) and has quite a few chapters on Israeli universities' teaching staff employment practices.