Sounds like a battery problem to me as well, but that's pure speculation on my part.
As "Huskerman" pointed out, the C5 is temperamental about batteries. It wouldn't hurt to
review the following thread (in particular, the videos it references) to ensure you've gotten the correct replacement battery for your car. As I recall, the battery model brought up in the videos was the side-post battery; I don't recall what year Chevy switched over to the top-post battery in the C5, but you may want to focus on the battery specifications rather than the model number.
Another point to make: if you let your C5 sit for long periods after infrequently driving it for only short distances, that's almost a guarantee that the battery life will be diminished. If yours (like mine) is mainly a 'garage queen,' you should probably invest in a good trickle charger. I personally use the
BatteryMINDer Model 12117, but others on the forum use the Battery Tender. There are plenty to choose from, all do essentially the same thing, and will keep your battery strong!
Another possibility is that you got a faulty battery earlier this year when you swapped it out. Have the battery tested; it could have a bad cell.
Good luck!