If the new scutes look good and the shell looks healthy, I'd say his shedding is a very good thing.
Is the other turtle also two years old and is the shell also two inches long? If so, both of them are small for their ages.
They need vegetable matter, yes, but since they're small and it sounds like they might not have had the best diet before, a good, balanced turtle pellet is also important (check out the Reptomin when you get a chance---I think it might be cheaper than the HBH pellets, especially if you get it at Walmart).
I give my turtles carrots about once a week. The mini carrots would work but borrow your mom's vegetable peeler and cut thin strips of the carrots-make them look like worms. Carrots last longer in the fridge than leafy greens do.
Red leaf and romaine lettuce last about a week for me. If you buy some and it's wet, it won't last as long as if it hasn't been sprayed with water in the store (I dry it with paper towels and put it in a fresh plastic bag--take a couple of extra bags---to make it last longer). If it gets wilted, let it sit in water to perk it up again. The anachris (comes in a bunch) will be fine if you just keep it in a container of clean water and feed a stalk to your turtles once or twice a week as part of the diet.
When you go the the supermarket for the lettuce and carrots, look around and see if they have an area where they sell fresh veggies at reduced prices (they're usually very ripe and won't last long). I always look for this section and often am able to get foods like red bell peppers and mangoes much more cheaply than paying regular prices. Both are OK for your turtles. You could also ask a manager (very nicely, of course) if they have any leaves of lettuce, etc., that they've trimmed off the produce (you don't need a lot). It's amazing what you can get if you tell people you want the food for turtles.

(If you do this and get some leaves, wash them well before giving them to your turtles.) Hope this helps.
"You become responsible, forever, for what you have tamed." -Antoine de Saint Exupery-