Yeah, my fish flip when I add plants to their tanks. In happiness! The only way you can kill fish by adding plants is if the plants are dead (or if they aren't aquarium plants, and are poisonous). Then they'd mess up your water and possibly kill the fish. But generally you notice the plant is going south before it affects water quality enough for that. Just avoid adding brand-new plants right before a vacation if you can.
Water sprite and frogbit are perfectly fine. Trim off the whole yellowed leaf. Anubias doesn't need root tabs, but some other plants will, so they're good to have on hand.
Flourish is an awesome fertilizer. :) With low light and Flourish, you can grow a nice variety of low-tech plants and not have to worry too much about them.
Examples of low-light plants you could grow now:
Most floating plants--frogbit, duckweed, etc.
Anubias--you shouldn't be having problems with this really, don't know why it isn't going for you. By the way, there are like twenty species of anubias easily available to aquarists if you order online--they vary quite a bit in appearance too.
Examples of what you can grow with Flourish:
Probably Creeping Charlie
Probably Ludwigia repens
Also, all your low-light plants will grow faster and do better with good fertilizer. Your Leaf Zone might be all right--I have no experience with it, so if I were you I'd take tankman's word that Flourish is better. Seachem is the trusted brand around here--they make Prime, Flourish, Excel (a liquid CO2 additive), specialty ferts in case plants have, say, a nitrogen deficiency, and even special substrate. What you need for your plants is Flourish (although if you don't have Prime, I highly recommend getting some).
Finally, if you get root tabs, you can grow crypts and swords too--but be aware that some of them DO have special needs (and, mainly in the case of the swords, some can grow too large for a 10g tank). Always research your plants before you buy, just like you'd research a new fish before you put it in your tank!