First of all, engineering training is substantially, if not almost entirely, the process of teaching the trainees a set of solutions to problems: what solvable cases are known, and how to apply the general approaches known to provide a solution in similar case, and how to apply them to your own specific case which require such a solution.
Sometimes these solutions have a good sound scientific explanations as to why they work, sometimes they are heuristics - rules of thumb.
This can very often lead engineers to believe deep down in their gut, at an unconscious level, that there are always solutions for everything.
Really, it's just that there have always been solutions for everything covered in their training.
Secondly, we are at our least creative when we are stressed and pressured. It sounds like you have felt this way for some time. This can lead you to "tunnel vision" in you approach. You feel the pressure, and the solution space that you are considering actually narrows. Most of the greatest "Aha!" moments in scientific and engineering history have popped into the minds of people when they were out strolling, or having baths, or, legend has it while sitting under an apple tree on a sunny summer's day. Einstein had his key breakthrough while travelling on a tram. This is because a lot of our cognition happens without explicit guidance from our conscious attention.
Given your current circumstances - pressure from the deadline, apparent insolubility, and feeling like you are carrying the weight of the company and the livelihoods of all the people in it on your shoulders - it is unsurprising that you are only able to bring half your brain to bear on the problem. The consciously attentive part.
Thirdly, talking issues over out loud brings different (and, therefore, additional!) neural pathways into play. It also forces each step of the problem-solving process, and each piece of information relevant to the problem, to be made explicit. Otherwise, your audience would not understand enough of the issue to contribute. Some of these required necessary steps and pieces of information get left unsaid when we think things to ourselves, in our internal dialogue.
This is a key advantage of being one of two or more people in a technical specialisation within an organisation. Other knowledgeable people that you can talk to. Failing that, find unknowledgeable people to talk to. Or a life-size cardboard cut-out of a famous fictional character. Or a rubber duck!
Many a problem in software development has been solved by what I was introduced to as the "deaf ear" technique. It has also been documented as Cardboard Programmer or Rubber Ducking at the software engineering patterns repository.
This is would be one of the key benefits of discussing this with your line manager, the senior management or your peers at your company. Or (with key proprietary details withheld) with friends and acquaintances from your days as a student.
4) Brainstorming is another technique that is relevant here, for jogging loose your additional brain capacity. The idea is to express all thoughts that occur to anyone in the group - event stupid ludicrous ones - as these can often jar unexpected new relevant productive thoughts loose in the other participants.
Perhaps the company should organise a company-wide brainstorming session. Maybe hire in some other engineers for the week that happen in. Two days bringing the new engineers up to speed. A day with a large group brainstorming as part of a facilitated joint session
Like many others here, I've started off dealing with this question as an XY problem. You're asking about how to resign, but we are all convinced you would prefer to stay if you could see a path to a technical solution.
5) To a man with a hammer, every problem looks like a nail. You're focussed completely on engineering your way to a solution given the current set of constraints on the approach. Involving more of the company in the process could help identify where the constraints could be adjusted, which would allow new approaches to become potentially useful. Perhaps the solution to the company's problem lies outside the engineering sphere, by moving the engineering target somehow.
6) If you and your managers and the company's senior management all decide that you have exhausted the other approaches outlined above, then it's time to move on.
You may have noticed that I, like many of the others responders, think this is an XY question. You're focussed on asking how to solve X, but really, you are trying to solve Y.
But, just in case:
7) In parallel with options 1 to 6, keep your eyes open for possible new opportunities in other organisations.
At which point, if you decide it's best, 8) get a new job offer and give notice that you are leaving.
I have gone from being one of a technical group, then moved to a different organisation where I was the only technician in my field. That situation is a lonely existence, and brings pressures and difficulties, especially to social animals like humans.
Good luck!
