Airlines publish different prices in different fare buckets. As soon as the cheaper fare buckets are sold out, the more expensive ones are automatically used. So to your first question, yes, the flights get more expensive when they fill up.
To your second question: Flight plans are usually planned for half a year, summer and winter. If a flight is full, then it's full. Usually an airline doesn't have the capacity to schedule another flight on short notice. They would have to hold replacement aircrafts and cockpit crews just for this rare case, which costs a lot of money.