You seem to have been given a written warning for rather dubious reasons and if there isn't a HR officer, do they actually have any HR procedures ?
I worked in HR a long time ago, so my knowledge is probably a little out of date, but... to issue written warnings they do have to follow Company procedure and you are entitled to appeal their decisions if you feel they have been issued unfairly. There's usually a 2 week window to appeal and the procedure is usually outlined in the letter and the HR manual.
Personally, If you are set on leaving, I would challenge the written warning, as it's on your record and potential new employers can ask about them. This would not only possibly clear your name, but it would buy you a bit of time between leaving as they will have to investigate your appeal.
I mean what are you incapable of exactly, have they offered some training to help you become more capable in your job ? If they're gonna hang you with written warnings, fire back with policies and best practise.
i think you are under the threshold to claim unfair dismissal, and it does sound a bit like they are trying to get shut. If you're not gonna get a decent reference and you've got a written warning on your file, I think I would be inclined to just resign and phone in sick for my notice period.