I definitely disagree about dropping during the game, that gives you an unfair advantage, and I've never personally noticed being allowed to do that in Yahoo, so that's surprising.
Basically, it doesn't look like espn did anything wrong here. You picked up a player that was available on waivers on Thursday because that's the day they were dropped. Our league has a mandatory one-day (full 24 hour period, so in this case Friday - Saturday) waiver period. According to the activity report, your waiver claim processed Saturday morning, as it was meant to. I'm not sure beyond that what may have happened that you either missed it, or something glitched on your end.
I'm confused about your complaint about the dolphins, since it appears you picked them up in free agency three weeks ago.