@rlhe,
First, who would like to check if the sentence is grammatically sound? It is not proper English to omit the subject of a sentence. In this case, it appears what you ought to have written is "I would like . . . "
The sentence is fine as it is written, and it is in the present indicative. However, it is nonsensical. You write: "While not entirely a sad song, “Hey Jude” is nevertheless as moving as it is uplifting."--which implies that there ought to be some aspect or aspects of the song which are sad, but that clearly is not what moving and uplifting mean. It would be better to make two sentences to express what you are attempting to say. For example, you might write:
" 'Hey Jude' is not at all a sad song. It is as moving as it is uplifting." There is no use for the word nevertheless in what you are expressing.
As a stylistic note, in American practice, when a sentence is enclosed in double quotation marks, any use of quotation marks within the sentence would use single quotation marks. (British usage is different, but those people are clumsy at best in their stylistic approach to the language.) So, i might write:
James stated to the police officer: "I heard a shout, and when i turned, is saw a man running, who said: 'That guy has a gun!,' so I began running myself.":