1). It is not advisable to pay up-fare, fees etc and hope to get reimbursed afterwards; with TK that is unlikely to happen when you can get a free (invol) rebooking due to Covid positive. Do the rebooking as soon as you can and hopefully TK will be able to speed things up when it is on short notice. You could also contact your local sales/ticket office for assistance; they might be able to help you out faster.
2) call TK CS or contact your local TK sales/ticket office. I have yet to hear of any positive coming out of trying to reach TK by Twitter or FB.