Unless you're actively, and rather loudly, suicidal... It's very difficult to get hospitalized in most places... Even if you want to be.
What the ER can do, and does very well, is clean and stitch you up. Mental health? Not so much. There's usually an on-call psych someone (doc, nurse, intern) around somewhere to do evals (and most evaluations end up with being sent home), but getting them down for a consult can take hours unless you're very lucky. So you'll probably have a 3rd year med student stitch you up, and IF you mention your anxiety is through the roof & is there anything they can do????Maaaaaybe a shot or a pill of something anti-anxiety related (they will almost never prescribe it, but will just dose you with it, and send you to your GP, as antianxiety meds are all super-addictive, they have drug seekers in also every night), if you're very-very-very up front about it. Either by outright asking, or by having a meltdown to the level of making it difficult for them to ignore you while they're trying to work on you & other patients. Really. Trauma Docs (ER) tend to have virtually no psych background, and their nights are filled with people running and screaming through the halls (hopefully, fully dressed, but not always). Their tolerance levels for "what's wrong with you" are very, very, high. Most psych problems don't even begin to blip their radar. You're not actively dying? You can walk & talk (and therefore are not really in that much pain)? You're fine. Let's stitch you up and send you home.
In point of fact? Ask the NURSE for antianxiety meds. ERs will start seeing so many people crashing under holiday stress that this is actually the easiest time of year to get a shot in the ass, or a pill to calm down on, but it's the nurses who generally make the recommendation to the doctor, as they're the ones dealing most with distraught patients (and can tell better who is faking, and who is seriously on the edge).
So jumping ahead to your kids being embarassed if you get admitted? Is the cart waaaaay out in front of the horse. Go get stitched up. Get you cut cleaned so it doesn't get infected and you end up losing an arm to gangrene or admitted for weeks with a blood infection. One thing at a time.