Getting rid of wild scallions
The most effective means of eradicating wild scallions (or wild onions) is by hand with a trowel because the small bulbous roots underground must be removed in their entirety to ensure the weed won't return.
If digging is impractical, either because the infestation is massive or because the task is too laborious, chemical herbicides can be used. Be sure to read the label to ensure the product is safe for use in vegetable gardens. Use of a "spreader sticker" ingredient is essential, as the plant's glossy stems will repel the chemical without it. Some herbicides already include such a surfactant, so be sure to check the label.
To further aid absorption, bruise the plants before spraying by stepping on them. That will open up a wound through which the herbicide can enter. Be aware that products such as Roundup Weed & Grass Killer will also harm surrounding plants, so if there is other vegetation in the area, you'll have to apply it to each individual weed with a small sponge. Steel wool would work double-duty, injuring the plant to allow for product absorption in the process. Be sure to wear rubber gloves if you do this. Though this is less strenuous than tilling, you'll likely have to repeat the process in November and again next year, as bulbs that are dormant this season spring to life.
What you've got there, Alan, is a classic case of cedar-apple rust (Gymnosporangium juniperi-virginianae), a fungal disease that targets junipers, as well as cedars, apples and roses. On junipers it causes swollen growths on branches that become horn-shaped, gelatinous and orange, as you've reported. Those growths, or galls, release spores over the course of several weeks in spring, which become airborne and can infect other plants. You're noticing them more during wet weather because all that moisture turns the gelatinous tendrils orange and causes them to swell. The good news is the disease is generally benign.
One way to avoid rust is to plant resistant varieties (check the plant tag before purchasing). It's too late for that, so your best bet would be to remove the galls in early spring and treat with a sulfur product, following label directions carefully.
Severe cases can be managed with a fungicide, but it's usually not necessary. If the situation does get to that point, look for a product that targets cedar-apple gall and is specifically intended for junipers.
I, too, have a crazy dog that trashes my garden. The only difference is that, in my house, I'm the one threatening my husband about the dog's numbered days. This year alone, Maddie has completely severed a 3-year-old jasmine vine that covered an arbor, dug up ornamental grasses, uprooted countless perennials, trampled tender seedlings and gnawed straight through a 2-inch-thick butterfly bush trunk. And don't get me started on the lawn.
There are a few dog repellents on the market, but I have no experience with them, so I can't really tell you whether they would be effective. One is Messina Wildlife Dog Stopper. I've tried their Squirrel Stopper product, and it's been effective in my bulb garden, but I can't vouch for the dog repellent.
What should work, if it's practical in the area you're trying to save, is a metal garbage can lid. Bury one just below the surface in the dog's preferred digging spot. She'll get that awful fingernails-on-a-chalkboard sensation when she tries to dig and, after several attempts, digging won't hold the same appeal. And you might try burying fully inflated balloons in the holes she's already dug. The next time she tries to dig, she'll get startled when her claws pop the balloon, and with any luck, after a few tries, will give up.
Good luck!