You can say "我三岁了" in a present tense from the day you turn three to the day before you turn four
At any age younger than three, you cannot say "我三岁了" without referring to the change of state is in the future e.g. 明年我三岁了
At any age older than three, you cannot say "我三岁了" without referring to the change of state was in the past e.g. 人類登陸月球時我已經三岁了，當然有看過直播 (I was already three years old when humans landed on the moon, of course I have watched the live broadcast, )
When people ask you: "how old are you" you can answer either: "我三岁了" (directly stating you are currently thee years old) or "我明天四岁了" (indirectly indicate you are currently still three years old)
"我三岁了，明天我四岁了" are two disjointed sentences
If you insist on stating the two facts (I am 3 already + I will be 4 tomorrow) in one sentence, you can do that, but only one 了 and one 我 is needed --> "我三岁，不過明天就四岁了" (I am three, but I will be four tomorrow).
The single 了 at the end applies to both facts (I am 3 already + I will be 4 tomorrow)