Every Lowes or Home Depot I've seen has a large panel saw and someone who knows how to use it.
When you buy your MDF or plywood, go on a week day. Those stores are usually too busy on the weekend, and maybe the guy who is good on the panel saw isn't there. Bring your diagram, and offer the guy at the panel saw $20 to cut the sheets. Whenever I've done that, they always refuse the tip (company policy?) but they do a good job on the panel saw.
It's much easier to take home those smaller pieces than a 4×8 sheet, and if the dimensions are not exactly what you need, you can clean that up at home.