这里存在着几个问题,不光是 learn / learning 的不同
看字面意思,想要表达的是“最近几年我一直忙着学英语”
(1) be 动词 用现在时 am 就不妥了,应该是 I have been busy with
(2) busy 后面不加 with 直接说 leaning English 比较合适；加 with 也行,但通常会说：I've been busy with my learning of the English ...这样才合语法和口语习惯
(3) these year 要复数,years,添个 few （these few years) 更好些