Both sentences are grammatically correct Dutch. The difference is emphasis, not grammar: "Helemaal alleen je eigen weg zoeken" puts the focus on helemaal alleen (completely alone), while "Je eigen weg helemaal alleen zoeken" is a more neutral statement where je eigen weg is the starting point. Dutch word order is flexible enough to allow both, and native speakers use either depending on what they want to stress.
The rule at play is the middle field (middelveld) order in Dutch main clauses. After the finite verb, adverbial phrases like helemaal alleen and objects like je eigen weg can appear in different orders. The default tendency is time, manner, place (tijd, manier, plaats), but this is a strong tendency, not a rigid law. When you move an element earlier, you give it more emphasis or treat it as known information. In your first sentence, helemaal alleen comes first, so it sounds more dramatic or contrastive. In your second, je eigen weg comes first, so the aloneness feels like an added detail.
There is no single "correct" order here. Both are idiomatic. If you want to sound neutral, put the object or the more concrete element first: Je eigen weg helemaal alleen zoeken. If you want to emphasize the isolation, put helemaal alleen first. Context and intonation decide which fits better.
A practical way to check yourself: ask what information is new or surprising. Dutch tends to place new, important information later in the sentence, and known or topical information earlier. If you are contrasting with someone who had help, helemaal alleen may deserve the early spot. If you are simply describing the act of finding your own path, the neutral order is safer.
Next steps: read Dutch sentences aloud and notice where the stress falls. Try rewriting the same idea with different orders and ask a native speaker which sounds more natural in a given context. For structured practice with Dutch word order, resources like Dutch Fluency can help, but any good grammar book or conversation partner will do. The key is to stop looking for one right answer and start noticing emphasis.