Combine apples, white sugar, light brown sugar, lemon juice, lemon zest, cinnamon, nutmeg, cloves, and allspice in a pot; bring to a rolling boil. Cook at a boil, stirring occasionally, until fruit is soft and jam gelling point (220 degrees F (104 degrees C) is reached, at least 45 minutes. Put a small amount of jam on a cold plate; freeze for several minutes. If the mixture is gelled, it is ready to process in a canner.

Sterilize the jars and lids in boiling water for at least 5 minutes. Pack apple jam into hot, sterilized jars, filling to within 1/4 inch of the top. Run a knife or thin spatula around the insides of the jars after they have been filled to remove any air bubbles. Wipe the rims of the jars with a moist paper towel to remove any food residue. Top with lids and screw on rings.

Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower jars into the boiling water using a holder. Leave a 2-inch space between the jars. Pour in more boiling water if necessary to bring the water level to at least 1 inch above the tops of the jars. Bring the water to a rolling boil, cover the pot, and process for 10 minutes.

Remove the jars from the stockpot and place onto a cloth-covered or wood surface, several inches apart, until cool. Once cool, press the top of each lid with a finger, ensuring that the seal is tight (lid does not move up or down at all). Store in a cool, dark area.