Why does this reward the cash and kill twice instead of once?

The problem is that the weapon is coming into contact with multiple parts of the player.

It looks like you added lines 4-32 to a script that already had the rest of it defined. ex, it looks like you're having the humanoid take damage twice: line 50 has a :TakeDamage() call, and line 19 has a health decrease (TakeDamage is preferred since it will respect forcefields). You should merge the 'if' statements. Note that there's already a "debounce" feature (which fixes your problem) used on line 47, if player and not HitPlayers[player] then -- if a player has been hit with the weapon, lines 48 - 53 won't get run again. You should have your "if humanoid is dead" code there.

[EDIT] "How do I merge 2 ifs together... And which lines do you recommend to be removed?" On line 4, you have if humanoid == nil then. You can combine the code you have there with the if block that starts on line 33, if humanoid ~= nil and MyHumanoid ~= nil and humanoid ~= MyHumanoid then. In other words, you need to move some code around:

function BlowDamage(hit, damage)
    local humanoid = hit.Parent:FindFirstChild('Humanoid')
    local player = PlayersService:GetPlayerFromCharacter(hit.Parent)
    if humanoid == nil then
        BLOCK1
    else
        BLOCK2
    end

--Delete the above "if humanoid == nil" if block and move the code blocks (marked by BLOCK1 and BLOCK2) to the below section:

    if humanoid ~= nil and MyHumanoid ~= nil and humanoid ~= MyHumanoid then
        if not MyPlayer.Neutral then        
            -- Ignore teammates hit
            if player and player ~= MyPlayer then
                return
            end
        end
        -- final check, make sure weapon is in-hand
        local rightArm = MyCharacter:FindFirstChild('Right Arm')
        if (rightArm ~= nil) then
            -- Check if the weld exists between the hand and the weapon
            local joint = rightArm:FindFirstChild('RightGrip')
            if (joint ~= nil and (joint.Part0 == Handle or joint.Part1 == Handle)) then
                -- Make sure you only hit them once per swing
                if player and not HitPlayers[player] then
                    TagHumanoid(humanoid, MyPlayer)
                    print("Sending " .. damage)
                    humanoid:TakeDamage(damage)
                    Handle.Splat.Volume = 1
                    Handle.Splat:Play()
                    HitPlayers[player] = true
		   BLOCK2 --put code block2 here, but take out duplicate lines (note how, in the previous few lines, a sound is played and the humanoid receives damage)
                end
            end
        end
    else --this line is new
	BLOCK1 --put code block1 here
    end
end