I find the Crawlers pretty easy to headshot, and the Knights seem to at least take a lot of damage if not one hit when their shields are down, if aiming at the head. The Watchers don't exactly seem to have a head, and focusing on the center "head thing" doesn't seem to improve the take down time. I'm finding them super hard to eliminate, and they often overwhelm me. Do they have a weak spot like other enemies?
Watchers do have a head. It's really small and just above the circular light on the chest. The Halopedia has a good close up of what they look like. I've been playing on a 1080p 42" screen and didn't know they even had heads or legs until I looked them up on the wiki. I've had Watchers go down after one or two well placed hits on occasion, so it just may be that the target is really hard to hit given its small size and their mobility.
One tip I can give is that, they usually fold up their wings and fly off when they are one shot from death. This is when you need to make sure and score another hit, or else they seem to come back in a few seconds at full health.
In my experience, the "weak point" of the Watcher isn't a particular point on it's body, but a weapon type.
Trying to hit any particular point on a Watcher is going to end in frustration. They move fast, there are terrible consequences for leaving them alive, and they've got a freedom of motion that no other enemy in the game really has. What you should be focusing on is killing them quickly, rather than shooting them in the most advantageous spot.
That said, my "trick" is automatic weapons. The Suppressor in particular will tear through a Watcher in seconds, although even the humble AR frequently does a better job than the mid-tier 1/3 shot burst weapons. Their wingspan is their undoing - it means that more of your rounds connect, even at ranges where typically an automatic weapon is less than effective. A single clip will tend to bring them down, even on the higher difficulties.
Not so much a weak point but one of the best ways to dispatch of a watcher is using a Needler.
Watchers are slow moving and not very good at dodging which is perfect for a homing weapon.
This only really applies to Spartan Ops where you have Covenant weapons to use against the Prometheans. This rarely happens during the campaign.
If you have good aim it takes 1-2 less headshots on the watchers. Use precision weapons to QUICKLY hit them in the head as much as possible. When the watchers fly away it will only take 1-2 more shots to take them down on any difficulty. The key to any enemy in Halo is the mid-tier weapons, they beat out any enemy faster. (If you are a crack shot)
Watchers have not real "Weak Spot" but are easily stun-able, like crawlers. A few shots from any weapon consecutively will kill them. I would recommend an Assault Rifle or persuasion weapon because of their movement speed and the time of the stun isn't that long. DO NOT let them flee, when they do, kill that one quick cause it will just become a pain later. They are support units that can actually make the difference between victory and defeat. Kill them first before engaging other promethians if possible.