Making the code you have better

Connecting events inside other event handlers is usually wrong. In this code, every time you click, you start another loop that waiting for Move and Button1Up events.

You refer to mouse.Target 5 times; just make a variable. (This also seems to be necessary in some strange situations because mouse.Target can be re-computed within a single frame)

dim should not be its own variable, it should be a part of target. The simplest way is to just make target nil whenever you don't have a target (since nil indicates absence this is very natural)

Don't use silly abbreviations like plr, just type the full word out, player. ROBLOX can help you autocomplete words and it's easier to read and type.

Don't use == false and == true, just use the plain boolean (with not if appropriate).

Clean code reads like well written prose

The key to writing good code is just writing exactly what you intend.

local player = game.Players.LocalPlayer
local mouse = player:GetMouse()

local target = nil -- initially, there is no target

-- RETURNS whether or not `point` is near the player
local function playerIsNear(point)
	assert(typeof(point) == "Vector3")
	local character = player.Character
	if not character then
		-- the player has no character, and so is near nothing
		return false
	end
	-- R15 doesn't have a Torso; use HumanoidRootPart
	-- for the "central" part of the player
	local root = character:FindFirstChild("HumanoidRootPart")
	if not root then
		-- the player has no root part (is probably dead)
		return false
	end
	return (root.Position - point).magnitude < 10
end

-- When the player clicks on an unlocked Union,
-- make that union the `target`
-- (noting whether or not it was anchored)
-- and anchor it
function onClick()
	-- When the player clicks...
	local union = mouse.Target
	if union then
		-- ... on a ...
		if union:IsA("UnionOperation") and not union.Locked then
			-- ...not-Locked Union ...
			if playerIsNear(union.Position) then
				-- ... that is near the player

				-- Make this the new union,
				-- noting whether or not it was anchored
				target = {
					part = union,
					wasAnchored = union.Anchored,
				}

				-- Anchor the union.
				union.Anchored = true
			end
		end
	end
end

-- When the player moves the mouse,
-- move the target (if any) to where they are pointing
function onMove()
	-- When the player moves the mouse,
	if target then
		-- If there is a target,
		-- Move the target part to where they are pointing
		local pointing = mouse.Hit.p
		target.part.CFrame = CFrame.new(pointing.x, target.part.Position.y, pointing.y)
	end
end

-- When the player releases the mouse button,
-- stop "targeting" any part (if you were)
-- and unanchor it (if it was originally unanchored)
function onRelease()
	if target then
		-- If the player was targeting something...

		-- Unanchor it (if it was originally unanchored)
		target.part.Anchored = target.wasAnchored

		-- Stop targeting the part
		target = nil
	end
end

-- Make the events fire when you want them to
mouse.Button1Down:Connect(onClick)
mouse.Button1Up:Connect(onRelease)
mouse.Move:Connect(onMove) -- Heartbeat may be better

Notice how the comments say exactly what you want to do? Notice how similar the code is to the comments?

Making the change

You want to change

When the player moves the mouse, move the target (if any) to where they are pointing

to be

When the player moves the mouse, move the target (if any) to where they are pointing IF THEY ARE POINTING TO SOMETHING NEAR

Well, just go to that part of the code and add it:

function onMove()
	-- When the player moves the mouse,
	if target then
		-- If there is a target,
		-- Move the target part to where they are pointing
		local pointing = mouse.Hit.p
		if playerIsNear(pointing) then
			-- ... only if they are pointing near themself
			target.part.CFrame = CFrame.new(pointing.x, target.part.Position.y, pointing.y)
		end
	end
end

Why is the code longer?

Your code is 31 lines (27 are not blank).

My code is 85 lines. But:

• 10 are blank lines (as opposed to 4 in yours)
• 28 are comments (as opposed to 0 in yours)
• 1 is an assert

Which only leaves 46 lines.

• 6 of which address error conditions your code doesn't (in playerIsNear)