The flavor and aroma (which boiled additions lack to some degree) is the goal of dry hopping. If you don't like the flavor, I'd say don't do it. The chunks being left in the beer is a problem with technique, though. It shouldn't be hard to keep them out of the bottles or keg if you care to.

The high alpha acid won't matter at all for late additions or dry hopping- only the other flavor and esters. Alpha acids have to isomerize by prolonged boiling in order to be soluble- and they only contribute to bittering, not as much to flavor and aroma. For the uses you describe, the alpha acids present are unimportant.

Bottom line: it sounds like you don't like the "grassy" flavor associated with dry hopping. Different hops taste different, and grassiness is more or less pronounced depending on the type. I would say: break the batch into two fermenters and dry hop each with different hops and compare later. However, if you don't want to potentially sacrifice enjoyment for science, go with the late boil additions and it will give you some of the hoppy flavor, but not so pronounced and grassy as dry hopping.

Cheers!