Apple Watch sending iMessages as iCloud email address..

Don't keep un-pairing the watch. That's a major step. Very time consuming and you lose data every time. Plus having it to set it up again from the backup. Odds are you're going to cause a major issue at some point. This really should only be done for a major issue.

With your Messages on the phone, I'm assuming you have "start new conversations with" set to your phone number. If you initiate a new message to someone from your watch does it go out as your number? Two things could be happening, you said that when you reply it goes out from your email address, that indicates to me that that is how it was sent to you. I run into this with my wife, sometimes her texts to me come in from her email address and if I reply back to that message then we just perpetuate the rest of the messages through her email address. So you can try to get the person who sent the original message to send another, new message and make certain that the person is sending to your phone number. Or delete all the previous messages from that person from the watch because if you reply from one of those it's going to go out again from your email address. Then send a new message from the watch to that person. That one should go out from your phone number, and any subsequent replies should also.

Most of these issues originate with the phone and while un-pairing can some times fix it ( temporarily ), that's the nuclear option. Try going to your phone's Settings/Messages and as long as you're certain all the settings there are correct, toggle off iMessages and then toggle it back on and try again. But make certain that you have deleted all the messages from your watch before any new conversations. Also just restarting the phone and watch every couple days can resolve some of these issues. But this is not one I've seen so far with my watch so I'm thinking you're replying to a message that came in to your email address. You could also uncheck your email address from the list of "Send and Receive" but I'd make that your last resort.

Hello

That part of the messages are setup when the Apple Watch is paired to the iPhone.

If that connection fails or gets out of sync,

First make sure the iPhone has the messages the way you want.

Unpair and repair the Apple Watch

That has worked for me.

Best of Luck

You made no mention that you had done that or any thing else. That very step, worked for me just 2 days ago, I didn't need to keep repairing.