"My leather shoes are old. I'd like to get a new pair."
In fact, there is a grammatical error in the Chinese translation. The subject in the first sentence is "我的皮鞋", and the subject in the second one is omitted. Hence, we consider the subject in the first sentence (我的皮鞋) as that in the second sentence. As a result, the sentence actually means "My leather shoes are old. My leather shoes would like to get a new pair." The solution is to add the subject "我" to the second sentence. "我的皮鞋很旧了, 我想买一双新的."
I am not sure if 了 is necessary as well, especially since the 很 is already there. There is no emphasis on the change of 'status' of the shoes to old, after all.
我想买新的一双 - I would like to buy a new pair. 我想买一双新的 - I would like to buy a pair of new ones.