She gave me a pencil. vs. She handed me a pencil.
She gave me a pencil could mean she gave me pencil as a Christmas gift. Or it could mean she left a spare pencil in my locker. On the other hand, she handed me the pencil indicates that we were face-to-face at the time of the transaction, and she put it in my hand. In other words, handed is more specific and descriptive.
She looked at the carrots. vs. She eyed the carrots.
I don't see these as equivalent, either. NOAD defines eyed as:
eye (v.) to look at with interest [emphasis added]
She eyed the carrots, then, is more vivid than she looked at the carrots. Perhaps she was walking through the farmer's market, and the carrots were a brilliant orange, in peak season, looking scrumptious enough to make her mouth water. If that were the case, she looked at the carrots would be a rather bland way to describe her fervency.
I supposed you could have picked alternate wordings to describe her actions and get around this problem:
She placed a pencil in my hand, and then ogled the carrots.
However, much like the commenters to your question, I'm at a loss to figure out why this body-part-as-verb phenomenon "irks" you. Moreover, as I mentioned in my comment, these verbs are centuries old; it's not like they represent some newfangled way of expressing something. And even when a body part is rarely used as a verb, I'm glad we have a language that is flexible enough to let it be used as one:
Orr's best remembered moment, of course, came when he wristed a Stanley Cup winning shot past Glenn Hall of St. Louis, then sailed gleefully through the air...1
It seems like if there would be any reason to be irked, it would be the other way around:
Sorry, you can't use that as a verb. It's a body part.
Now that would be vexing – and hard to enforce. I wouldn't want to be shouldered with that duty!