How to fix these 2 problems for my Battlefield script?

The first problem is caused by you not using a local variable.

When you say m =, it refers to some shared m -- one that any different call to this function will use.

That is almost never what you want, and modern coding practices say this is a very bad thing to do.

Make m local:

    ....
    local m = Instance.new("Message", game.Workspace)
    -- Parent is second parameter, makes it a little short
    ....

TL;DR: When first assigning to all variables, declare them as local (you can search on SH for lots of questions concerning how to use local)

The second thing would be a different sort of debounce. Instead of us using true and false, we'll use a dictionary:

announced[model]

will be true if model has been announced, and otherwise nil. We just need to use this instead of debounce:

announced = {} -- Define our dictionary

function announce(part) -- Use better variable names >:(
    local model = part.Parent
    if model and not announced[model] then
        announced[model] = true
        -- We want to make sure `part.Parent` isn't `nil` or nasty
        -- things happen (in addition to the debounce)
        local m = Instance.new("Message", game.Workspace)
        m.Text = model.Name .. " has entered the battlefield!"
        wait(3)
        m:Destroy()
    end
end

Because of how table indexing works, we can use model (as opposed to model.Name) to capture a reference to that particular model -- when you respawn, you'll get a new one, so it will ignore the old entry.

This is a neat trick that avoids us from having to track the .Died or .CharacterAdded events or anything else -- we get it for free!

(This will accumulate garbage; it will remember every player who has ever touched it in any life, even if they're long gone, so it might be a good idea to clear out values that are no longer in the Workspace, though this shouldn't be necessary)

To make sure it doesn't go when it already announced is to insert a value into their player and then check it. As for the not stopping, you never set the debounce to false - and never defined debounce as false.

local db = false

function check(plr)
    if plr:FindFirstChild('Touched') then
        return true
    else
        return false
    end
end

script.Parent.Touched:connect(function(hit)
    if db == true then db = false
        local plr = game.Players:GetPlayerFromCharacter(hit.Parent)
        if plr then
            if check(plr) then
                Instance.new('IntValue',plr)
                local m = Instance.new ("Message",workspace)
                m.Text =  "A new player has entered the battlefield! Name:"..plr.Parent.Name
                wait(3)
                m:Destroy()
            end
        end
        db = true
    end
end)