Giving things away doesn't necessarily remove them from your encumberance limit
 

Steps to reproduce:

1. Be close to weight limit
2. kill mob with something heavy (like 2h weapon)
3. loot mob, but keep heavy thing stuck to cursor
4. Note that you are encumbered
5. give 2h still in your hand to pet
6. Note that you are still encumbered

This seems to wear off when you next loot anything, and I don't know if it's just a client/visual issue, or you are REALLY encumbered and suffering the run slowdown and lower AGI or not.

I'll add to this, since it seems like a related "issue". You can hand your copper to a NPC/pet and cancel a trade - your weight goes down, until the next time you loot. And it's not just the weight display that's bugged - I used the trick to get to a vendor faster on my alt who had copper in the thousands, so it does affect your speed. So it comes in handy every now and then, but it doesn't seem like an intended feature.

reported
look up the definition of exploiting

ex·ploit
verb
gerund or present participle: exploiting
ikˈsploit/
make full use of and derive benefit from (a resource).
"500 companies sprang up to exploit this new technology"
synonyms: utilize, harness, use, make use of, turn/put to good use, make the most of, capitalize on, benefit from; informalcash in on
"we should exploit this new technology"
"chewie exploited the shit out of the HT refresh timer running while offline"

You can chalk up that vendor run as a test of whether the weight updates back on its own, or if I had to loot a new mob to do it. I was doing it for you and for the rest of the P1999 community, to improve your gaming experience. Be grateful.