I also have family there, so I know this drill.
My best advice is to forget summer. It is much cheaper to go in Fall or Spring, and cheapest of all to go in January. IME, schools tend to be more flexible about family-related overseas travel than they are about domestic vacations, so long as you don't make an annual habit of it. If you can piggyback a week out of school onto something like spring break, you'll do much better on price. If you absolutely HAVE to go in summer, plan to leave the day the kids get out of school; late May into early June is your best bet.
Other than season, the two biggest issues are the fuel surcharge and the UK passenger taxes. The fuel surcharge is sometimes included in the fare itself and sometimes listed separately, and some airlines charge a flat rate and others vary it by fare class: make it a point to read the fine print.
The UK passenger taxes on int'l flights are the highest in the world (and are going up 55% this next April), so even though it is a PITA with kids in tow, it can sometimes pay big to fly into/out of Ireland and then take a separately-ticketed domestic flight (or the ferry!) to the UK. Oh, I just saw that you are planning to go to DLP; in that case fly in an out of France to save the taxes.
Also, is there any chance that Charlotte is a possibility for your departure point? Knoxville should be ruled out; the fares and extra travel time from a small domestic-only airport will bury you.
NOTE that if you are trying to save taxes by landing outside the UK, a simple connecting flight via someplace like CDG won't do it. The end point of the itinerary must not be in the UK, so it has to be done with a completely separate ticket.