If you have a 5 gallon bucket of water and a 1 inch hole, you can dump the entire bucket of water onto the hole, but the hole will only take so much water.
The chargers are the bucket of water.
The battery is the hole.
When current is supplied is causes voltage to rise. More current, more voltage. Once voltage gets to the programmed absorption voltage, all chargers will see this and then send only enough current to maintain that voltage. In the analogy, when the battery can't take more, the "hole" gets smaller.
Consider that each of the 120VAC outlets in a house can deliver as much as 2000W of power, or 15-20A of current. Why aren't you worried about plugging a 13W LED using only 0.1A into the power socket when the socket can deliver 100X that current?
The battery "talks" back to the chargers via its voltage, and the chargers respond accordingly. It's your responsibility to ensure all charge sources are programmed for your battery.
Unfortunately, Everstarts are pretty much junk. Count on them not lasting the two years unless you are VERY gentle to them with only slight discharges. They aren't really "deep cycle" batteries. They're more typical of starter batteries and have a poor cycle life. If you got the ones rated for "122Ah" note that is only for a 1A discharge. They are closer to 60-80Ah when you look at a 20 hour (standard) discharge rate.
You need to check their electrolyte levels frequently.
While it's not directly applicable to your batteries, the following two resources are a good education on flooded acid batteries, their opration and their maintenance:
The key to achieving optimum performance and long battery life is to follow a regular care and maintenance program. Read our tips for high performance battery maintenance.
www.trojanbattery.com