Sally's carries a number of grey cover choices... sprays, mascara to wand on color, a stick that looks like a lipstick, tiny powders with a brush (they look like eye shadow compacts) & some women even use temporary rinses (put on after shampooing), Roux & another brand. They even carry an expensive product (T o p i x... I think it was $23 per small container), which is fibers you sprinkle into hair... everyone I knew who used it loved it, as long as it matches your hair color.
I don't know which would be best for you... depends on color of hair, thickness/texture/density of hair, ease of use & which method you personally prefer, so it will be trial & error. For example... Roux rinses will run if caught in the rain or you jump into a pool. Good to know if wearing a white top & there's a summer shower, so you may want another method that will stay put easier if you're the outdoorsy type or live in a rainy locale.
My point in mentioning it is they don't have in-store testers, but do accept returns for 90-days with a receipt. So, you could buy a couple of them & test at home. In addition, don't be afraid to use 2 at a time, like with foundations where you may need to mix & match. You may find the most success with using the rinse, then when hair is dry, powder over it, or the spray touched up with mascara, etc. The biggest complaint I heard was not enough color choices being available... there are more color choices in the med-brown to dark-brown & black stages. I suppose it also depends on how much grey you need to cover... would be time consuming to use a small brush & powder all over, but if you're covering some strays in the front/temples, that mascara wand might be best.
I know Ulta also has some products, but don't know which ones or if they have in-store testers. You may want to look on their website first before venturing out to the store... at least where I worked for 1-mo, Ulta customers came in on the regular as they were all out. I'm unsure of their return policy, but I'd imagine they'd allow a return with receipt. While I'm mentioning them, also check Sephora... they might have some products, too.
Some women also add highlights & lowlights (done every 3-mos, rather than 1 flat color once/wk, as you mentioned) so that the gray just seems to blend in. Again, it depends on original color of hair & amount of grey. I'm a former hairdresser & sometimes making a coffee paste mixed into conditioner did the trick. Tea dying is another solution... again, depending on your own situation. My point is, if you have something at home that's non-toxic (eyelash mascara, eye shadow, etc), for the heck of it, test it on a couple of strands to see if it works for you. It might not be for everyday, but can hold you over 'til you find the perfect solution.
Oh & 1 tip. To make it less messy & frustrating, whatever you try, do 1 small section of your hair only (strand test). You don't need to cover every single grey hair to determine a product will work or not. If the color matches & you're going to cover the rest, do another test first & that is to use a spray bottle ($1-3 in the Sally's bins in the back of the stores or any Dollar Store) & spray that area with water to see if it runs, flakes, etc. I'd imagine they will all "melt" in rain (as does makeup), so maybe always carry a dark baseball cap/beret, just in case you get caught... they scrunch up nicely in any purse.
If you have other q's, would you mind posting some descriptives about your hair... original color and/or whatever color you currently color your hair, amount of gray (salt & pepper, 50% gray, temples only), hair density (thick/waxy, thin/fragile), straight/curly/wavy. It's more helpful with making recommendations or giving some guidance.
Good luck. I'm sure you'll find something perfect after experimenting a bit.
Please post back with your results as I'm sure plenty of others could use your info as a base to test the same.