Mentre in Inghilterra scrivo su LangCorrect website, ascolto podcast s e RAI Radio Due, anche lavor ando un libro.
In English, you can say “While…” not followed by a verb. In Italian, you have to specify that verb. Also, we usually use the word “quando” in this context.
Website = Sito (web)/Pagina (web).
We don't add an 's' to form the plural if we use an English word (I have two computers = Ho due computer).
In Italian, if you say “anche” at the beginning of a sentence (main or subordinate), it means “too” and it implies a different subject:
A: Parlo italiano (I speak Italian)
B: Anch'io! (Me too!)
In this context, since I've taken a look at the original text, and you wanted to say “as well as”, you can say “così come”. However, it doesn't really work in this case, because you'd have to change the verb tense, so you can just say “e lavoro...”.
The verb “lavorare” without prepositions means “to work”, just like in English. We don't really have a direct translation of “work through” in this context, so you can just say “Lavorare con un libro di testo”. Speaking of which, textbook = libro di testo. If you just say “libro”, it means “book”, just like in English.