The concept of "Too much" does exist in Japanese, however, I can think of different manners of expressing it. In your question you ask about "Being too much", but the most common pattern, in my opinion is "To do something in excess" that is expressed as:
V-ますstem + 過{す}ぎる
To answer your question, I will first point to your example. What you say is correct, albeit verbose. I would go with the shorter:
もうたくさん、食べられない
(see weblio, second sense)
You have another option which is to use 余{あま}る
or one of its compounds. By itself, 余る can mean "To be in excess". However, I find a lot of examples using either:
手に余る : この仕事{しごと}は私{わたし}の手{て}に余る。
Which means "I can't handle it", "It's too much for me". Useful compounds are 余裕{よゆう}
, 余剰{よじょう}
which both have a meaning of surplus, excess. However, their use varies, and I can only encourage you to look them up in a dictionary and check for usages on the web!