My last answer was removed; apparently folks thought I was joking. So I'll drop the levity...
If you don't want to look like a "freak" (your word), then you want to be very careful about what you use to remove hair from your face. The skin on your face is extremely sensitive; while you can easily use many different tools and techniques to remove hair from other parts of your body, careless treatment of your face is nearly certain to result in damage to your skin that will look (and feel!) worse than the hair you started with. In particular, if you are prone to ingrown hairs you'll want to avoid cutting or breaking the hair at or below the skin.
In short, I'm saying you should avoid hacky solutions here. No matter how negatively you expect folks to react to you showing up with a day's growth of facial hair, showing up looking like you stepped out of the burn unit into a nest of wasps isn't going to be an improvement.
That said... You can certainly mitigate the impact of shaver breakage. Here are a couple of suggestions:
Top-down + bottom-up shaving
Like most of us, you probably shave by starting on one side of your face and then moving to the other, to allow you to clearly see what you're doing in a mirror by turning your head.
Well, stop doing that.
- Start at the top and work in narrow rows across your entire face down to your jaw line, being careful to still shave with the grain.
- Then start at the bottom of your neck and shave in rows up to your jaw line, again being careful to respect the grain.
- Finish your jaw line, and do a final pass perpendicular to the grain to catch stray hairs that may have been flattened during the first pass.
At no time during this process do you look like Two-Face. The worst that can happen is that you're left with a slightly odd-shaped band of stubble - and there's a way to mitigate that too...
Invest in a small, inexpensive electric clipper. These are essentially a miniature version of the ones your barber uses to trim your hair, and you can probably pick these up for $10-$12 at your local discount department store. You can even find packages that come with fairly crappy electric shaver heads as well, which has the added bonus of giving you a spare shaver and potentially side-stepping this whole issue.
Once you have your trimmers, use those to do a first pass on your face. Don't use a guard - just hold the flat head of the trimmer parallel to your skin, allowing the hair to enter the cutting blades without bringing the cutting surfaces themselves into contact with your skin. Note that the design of this tool is essentially a row of tiny scissors - which should thus comply with your religious obligations.
Once you're done, you'll be left with fairly short stubble. If your shaver breaks immediately, you could just go with that look - it's actually quite popular these days.
Assuming you want a closer shave, just finish it up with your normal shaver, using the top-down/bottom-up method described above.
The advantages of this method are threefold:
- The shorter hair will actually be easier for most electric shavers to handle without getting clogged, thus making it less likely that your shaver will break mid-way.
- There's less of a difference in hair length between the area you're shaving and the area you've already shaved - so if your shaver does break mid-way, you're not going to look as odd.
- As the trimmers can readily handle hair of any length, you should need fewer passes to complete the job - thus reducing skin irritation.