There's a more standard, general rule in English grammar that web searches tell me is called "delayed right constituent coordination". It lets you read sentences like "He washed and dried the clothes" as "He washed [the clothes] and dried the clothes." The same object gets applied to both verbs.

I suspect that's what you're applying to these sentences. "Try and finish the assignment" makes some sense under this rule if you read it as "Try [the assignment,] and finish the assignment" -- an "assignment" is a thing that makes sense to "try". ("He tried [sushi,] and liked sushi" works for the same reason.) But "Try [the truth,] and tell the truth" doesn't work -- it doesn't make sense to interpret "trying" the truth as some separate action you're taking before you "tell" it.

So probably you just don't have the article's special try-and "pseudo-coordination" rule in your dialect.

This makes a lot of sense, and it definitely explains where other "try and" sentences work while "try and finish" doesn't at all.