I am almost done my game, and will probably start over again. What I need to know is do the purchases of doubling tech points and doubling food still apply to a new game if I choose to start over from scratch. Or would I have to purchase them again for another $0.99 each?

When you use the Start Over button on the game, it will delete everything in your game, including villagers, puzzles, tech points and levels, food, and in-app purchases. However, the two "doublers" (for food and tech points) are a type of in-app purchase that can be downloaded again for free. Note that this is only possible for these two items. Tech point and food purchases cannot be downloaded again for free, due to Apple's guidelines for different types of in-app purchases.

To get your two doublers back: After you start over, make sure that you are signed into the App Store with the Apple ID you originally used to purchase them. Then you can get those two doublers back by choosing to "purchase" them again from the Upgrades section of the game. Since you have already purchased them with that account, you will not be charged a second time.