Two Astable NE555 Affects one another
Sounds to me like you have not decoupled the two systems properly. What decoupling capacitors do you have in place? And, equally important, where are they?

Based on what you've said there, you should have at least a 100nF across the power pins of each chip, as physically close to each IC as possible. I would also say that a large bulk decoupler (say 10uF) would be a good idea to fully stabilise them, you could put a single cap next to the battery.

No doubt someone with more time will be able to write a good detailed description of decoupling practices, but a quick internet search for "decoupling practices" would tell you a lot, as would "why do we need to decouple ICs?"