Agree with previous posts.
You will need to use fertilizer, as the 17 plant nutrients are unlikely to be present just from fish food, organics and tap water. I second Kymmie on the Flourish Comprehensive, I use it. Make sure it is Flourish Comprehensive Supplement for the Planted Aquarium
as they make several products in the Flourish line, this is the complete one and all you need for your setup. Without this I would expect your swords to start yellowing and not last, and cabomba being a stem plant is a fast grower (actually swords are pretty fast too) and thus requires nutrients to balance the light or it won't grow.
And don't be worried if your sword loses a few leaves (they will yellow), this is natural when the plant is moved to a new aquarium. As long as new leaf growth is emerging from the centre of the crown, the plant is probably OK.
On the light, I also suggest new tubes, and the hardware store is fine. Select tubes around 6500K (that's the kelvin rating which determines the colour of light emitted). This is closest to mid-day sun, and provides the red and blue the plants need plus green to balance for a natural colour rendition of fish and plants. Sylvania, Phillips and GE all make tubes like this, calling them daylight or daylight deluxe or something.
With two tubes over a 60g you should not have trouble with cabomba, which does require a bit more light as Kymmie mentioned.