Review the uses of na and compare this with nang.
Recall that na, among other uses, denotes completion of an action. Nang connects a verb with an adverb. In your sentences, the appropriate word is nang since the verb maneho (to drive) is modified by the word lasing (drunk), that is, nagmaneho nang lasing (driving drunk).
Colloquially, people will use na, nang and ng interchangeably in sentences like you have provided. This is especially true with nang and ng which are phonetically identical.
That said, you wish to highlight certain facts in each of your three sentence above. The syntax will be different in Tagalog.
1) Nagmaneho nang lasing si Tom. = Tom drove drunk.
2) Habang lasing, nagmaneho si Tom. = Drunk, Tom drove.
3) Lasing si Tom nang nagmaneho siya OR Si Tom na lasing ("Drunk Tom") ay nagmaneho. = Tom, who was drunk, drove.
Thank you, DotterKat. My native speaker rejects "nagmaneho nang lasing" - insisting on "nagmaneho na lasing" on the grounds that "lasing" does not describe the manner of the action. This seems more than simply a colloquial preference for "na". However, I find support for "nang lasing" on the internet, e.g.: Hindi siya puedeng mag-maneho nang lasing.
Pano ka nakapag-manejo nang lasing?
... so perhaps it is just that different speakers have different preferences.
Schachter & Otanes 1972 (6.7, 6.10) seem to accept "nang" only before an adverb of manner or time. In English, the Cambridge grammar analyses "drunk" in "drive drunk" as a "predicative" (or "stative") adjunct, since it describes the state of the subject while driving rather than the manner or time of his driving. In S&O the closest relevant example of "lasing" that I could find uses "na": Inabutan ko si David na lasing = I found David drunk.
and likewise Kroeger 1973: naghain na lasing si Maria ng isda = Maria served the fish drunk.
That said, I do see an advantage in consistently using "nang" before "lasing" because it would resolve the ambiguity of: nagmaneho si Tom na lasing (Is it drunken Tom driving or Tom driving drunk?)
I see that your last example cleverly skirted the ambiguity by using "ay": si Tom na lasing ("Drunk Tom") ay nagmaneho = Tom, who was drunk, drove.
Regarding (2), my native speaker agrees with the comma but still prefers "nang" or "na" after it: [Habang] lasing, [na/nang] nagmaneho si Tom. = Drunk, Tom drove.
..regardless of the presence of "habang". Again, I do not know if this preference is idiosyncratic.