Your dealing with a right set of nasty barstewards. You are doing the right thing by leaving. Basically, you owe them nowt - so leave yourself on sick pay covered by a doctor's note until either your notice time expires, or you have a conformed offer in writing of another job.
If you need to attend an interview during the time you are receiving sick pay, let your employer know via a simple letter thata you will attned a job interview for X hours on X date. Then they can deduct as appropriate or post it where the sun don't shone.
But do not give them an excuse to get at at you - so everything is formal; don't rely on phone messages or email; be covered by your do at all times.
If you get an offer and accept it, then you can write your letter of resignation to take effect as of the date you start your new job (or rather the day before...)
But f your new job is to start after the time that your notice runs out, what you do is bugger all to do with them.