On the basis of what you say above, the Company has not followed its own procedures. Since following the defined procedure is important, this could have helped you establish that the process wasn't fir in the event that they decided to dismiss you and you took the whole thing to tribunal.
They didn't do that - they gave you a Final Written Warning. Unfortunately by signing what you did, you've gone someway to acknowledging that you accept what they are saying, and hence you are accepting that the outcome is reasonable. However you could always later say that you did it under duress.
A Final Written Warning is basically what it says - it's our last chance, so keep your nose clean for the period that the warning lasts. Yes, you could appeal the FWW - I would suggest you do it on the basis that they failed to follow the correct process - not the decision.