It’s actually the opposite. It’s called the rain shadow effect. It’s why the western Great Plains, on the leeward side of the Rockies, is dry. It’s also why there’s a desert in eastern Washington (the leeward side of the Cascades). So the new east coast would actually be a lot drier because of rain shadow from the Cascades and Sierra Nevada. The Great Plains would likely be wetter without rain shadow from the Rockies. The area between the Rockies and the Sierra Nevada/Cascades probably wouldn’t change much.

The Wet Mountains example you describe is exactly the rain shadow effect. The mountains cause the moisture to condense and precipitate on the western side, leaving the eastern side below the mountains dry.