Learn German: Ich wollte sowieso nach Hause gehen.

Yes, absolutely! While sowieso means "anyway" or "in any case," B2 learners should also recognize its common synonyms. The word eh is a very frequent, highly colloquial alternative used in everyday speech (Ich wollte eh nach Hause gehen). ohnehin means the exact same thing but sounds slightly more formal.

Since this is likely a spoken sentence, shouldn't the past tense of wollen be in the Perfekt?

Not for modal verbs! While spoken German heavily favors the Perfekt for most verbs, modal verbs like wollen, können, and müssen are exceptions. Germans overwhelmingly prefer the Präteritum (simple past) form wollte here. Using the Perfekt with a second verb creates a clunky "double infinitive" (Ich habe sowieso nach Hause gehen wollen), which native speakers usually avoid.

Even at an intermediate level, it's easy to mix up nach Hause and zu Hause. Why do we use nach here?

This is a classic slip-up that persists even at higher levels. The rule to remember is that nach Hause shows direction toward a destination, whereas zu Hause shows a fixed location. Because the verb is gehen (indicating movement toward your home), you must use nach Hause. You would use zu Hause if you were already there and not moving, such as Ich wollte zu Hause bleiben (I wanted to stay at home).

Could I put sowieso at the very end of the sentence?