Read the whole thread thus far and watched your video. First off, I want to say that I am very glad you decided to come ask for suggestions here instead of heading out to Circle K. I feel like we're still missing some pieces that could solve this for you.
I have a bit for you to consider, mostly points I didn't see covered in the other replies thus far (sorry if I missed any posts and info repeats itself here), suggestions, and questions:
1.
Juice
You comment in the video that the ejuice is very thick. It does look thick, thicker than my 100% VG juice and as thick as my straight vegetable glycerine. Instead of modifying your devices, try adding a couple drops of distilled water to your juice to thin it out. If it's not wicking properly then modifying your clearo will not help, so far as I see it. The liquid is not travelling up the wicks properly because it's so thick, so what wick will be the lighter weight portion leaving even thicker juice behind. Once the juice is a bit thinner due to distilled water it should start wicking right; then you can start playing with the battery voltage and coil resistance to find the best flavour. Flavours come out differently at different voltage/resistance combinations, even burnt tasting if too hot an equation. If it tastes burnt, turn the Twist down.
I checked the ejuice you're using and it appears to be advertised as a PG juice on their site. I've not seen a PG juice as thick as it looks in your video which makes me suspicious; I don't have experience with altsmoke juices though. Either way, distilled water introduced drop by drop should thin it out. You may wish to do this either in a separate small bottle or just in a syringe (draw some air into the syringe after the juice and water, then shake to mix).
2.
Tipping
You shake your device a lot in the video. When we talk about tipping, we are trying to describe a gentle tipping motion. Shaking the battery may worsen a short/loose wire if there is one in the battery. YMMV but I don't carry my top coil clearos in my purse or pocket any more because of the potential leaking when they get shaken. I also check my battery for juice on the connection every few hours when using those clearos and wipe with a q-tip as needed.
3.
Flashing
If you press the eGo Twist button enough times it will flash. If there is a short circuit it will flash repeatedly and shut down. Is this the batteries fault or the clearomizers fault that yours is flashing? Check all your posts and be sure the battery is locked off before you fiddle with the post at all. You say your wife vapes; perhaps your wife has a 510 threaded carto or clearo filled with her own ejuice that you can test to see if the problem originates in the particular clearo you are currently using.
If what happened was a short then the Twist is doing what it is supposed to by flashing and shutting off the battery. This is good. You didn't just have a battery explode in your face. Score!
4.
Power
You received your Twist 2 days ago. Have you charged it since then? How have you charged it? I have a 1000mah Twist and find that it needs charging nightly when using it as my main battery since I like to use it in it's upper range of voltage. The higher the voltage you use the faster it will use up it's energy and need charging. It doesn't seem like this is what the issue is but thought it worth mentioning.
5.
Quality
It is entirely within reason that you may have a faulty battery and/or a faulty clearo and somehow your PG ejuice is way thick when it's not meant to be. Quality control is not perfect, even in the most wonderful manufacturer/retailer's stock. Just as there is a chance you'll hit jackpot on a slot machine, there is also the chance that at some point in your shopping you will receive faulty merchandise.
I have a black Twist and it seems like the dial at the bottom of yours has some kind of space between it and the tube. It should be a tight near flush fit. Hard to see in the video.
If your Twist is faulty then I imagine the site you bought it from will take it back. I'd suggest contacting them directly, including a link to the video you posted.
6.
Patience
Not just for reading my novella here!

I have had clearomizers that I unwrapped, filled, then tried vaping immediately that just refused to vape. After some frustration, instead of tossing them I now leave them overnight, filled with juice, then try the next day. Some cartos and clearos may also take up to a day to really get going and you have to almost baby them through it. I don't have this often but it happens. I've also had clearos that would play the game you seem to be describing: they will wick a tiny bit then taste burnt. Usually I find that those ones are closer to DOA than the aforementioned type that do nothing then vape fine the next day.
So be a little patient with new gear. It sucks, but it's the same as when you buy a handful of lighters for a roadtrip and discover a day into the drive that each one only lasts a few flicks. Does the Circle K carry the disposable cigarette shaped ecigs? Maybe pick one up as backup. Or ask the wife for a boost in that dept if she has her own gear. This can be figured out.