double-blind study - an experimental procedure in which neither the subjects of the experiment nor the persons administering the experiment know the critical aspects of the experiment; a double-blind procedure is used to guard against both experimenter bias and placebo effects.
No they get more expensive, airlines will always advise you to book in advance to take advantage of their best fares. Sometimes they do a special deal but not within a week of flying. I would try another airline or make different plans.
In most cases, the actual fare will do nothing but increase closer in to travel date. So yes, your best bet may be to grab the seats now. As you mentioned, this flight typically has empty seats. Often, an airline will float out it premium fare tickets in the hopes they will be grabbed up early.