The boat lies half-sunken in the bay.
Sunken is an adjective, right? But in the previous sentence, it seems to be acting as adverb modifying lies.
Should the sentence be:
The boat lies half sunk in the bay.
You seem to be asking two different questions.
The first is how an adjective like half-sunken can apply to a verb in your sentence. The answer is that it doesn’t, because lie is here functioning more like a copula. It just serves to link the subject with a predicate description of that subject. Some writers prefer the term “semi-copula” for the verbs other than be that work this way, especially since many of them can also work as non-copulas. But here are similar examples of verbs that take (or can take) adjectives in their complements:
So the use in your cited sentence is perfectly correct as an adjective complement of lie.
Your second question appears to revolve around whether the past participle of sink is sunken or sunk. The answer is that either will serve. Like many strong verbs, sink has enjoyed a range of possible historical inflections, and there still remains some ambiguity today.
In general, sink inflects like drink and shrink, which all enjoy, or can enjoy, a four-fold paradigm:
Where the fourth form is reserved for adjectival uses, as in a drunken man, a sunken ship, or a shrunken head.
That’s the way I myself personally inflect those verbs, but that’s just one model, and different speakers say different things at different times. In one common alternate model, we see this progression instead, which lacks an -ank form in the preterite and does not distinguish verbal from adjectival uses of the last form:
For drink, that alternate model is now considered non-standard, but for the other two, the case is less clear, and both models have some currency and acceptance. Think of the movie title Honey, I Shrunk the Kids, or the old boardgame’s commercial’s “You sunk my battleship!” That shows that the alternate model still lives for shrink and sink, even though saying “I *drunk a lot last night” is now considered thoroughly non-standard, as too is “I hadn’t *drank much that night.”
For sink, the OED gives the past tense as either sank or sunk, and the past participle as either sunk or sunken. It gives sinken as a vestigial dialect use of an older form, and reports that the weak inflection sinked was also sometimes found, albeit rarely except in one now-obsolete circumstance.
In some speakers, there may be a difference in transitive and intransitive uses:
The OED states that ‘In present usage [sunk] in adjectival use tends to be restricted to senses implying deliberate human agency’.
As we don’t know that that is the case in your example, sunken seems appropriate.