Hey Tony, first of all welcome to the club!
I'm sure there will be a lot of advice coming, I've never "smoked" boneless skinless chicken breasts, but here's my two cents...
Hickory and paprika do go together, however I've never heard of just using paprika by its self, I'd think you would want to blend it with other spices (things like salt, pepper, garlic powder, onion powder, and chili powder would be a good start. So just paprika could be the after taste, but I'm not sure.
Four chunks of hickory for chicken breasts does sound like a lot, and I'm one of the people who like a heavier smoke. But some people do find too heavy of a smoke off putting, and that could lead to an off taste. Hickory does have a strong flavor, fruit woods, like apple or cherry, are gonna be a lot more mild.
Another thing to consider is having a clean burning fire. You said you adjusted the vents to bring the temp down, do you remember how they were set? You want to keep a positive air flow, where the top vent is more open than the bottom vents. That's going to let you fire breathe or exhaust properly. If the top is closed more than the bottom you're gonna get a build up of "stale" smoke, and that would definitely create an off taste.
There's a lot of things to consider, so maybe some more details about you cook, vent settings, spices used, etc...
Again, just my initial ideas, the great thing about grilling is you get to learn and experiment, and this is a great forum where plenty of people will offer advice and tips! Keep it up, and don't be afraid to ask questions!