K.P.
It is not okay to lie about your residence for several reasons. First, accessing a program for community residents when you do not live there is not only lying, but it is also stealing. Second, you will also be asking your children to lie. If the preschool teacher wants to do a lesson on safety and they ask each child to bring in their phone number and address to "learn"- what will you do? You are sending a bad message and setting a precedent that your children are old enough to pick-up on.
If you cannot afford childcare and tuition, then this is not the right time to go back to school. Private schools do not often barter, so I would not count on that as an option. Based on their ages, if you wait two more years they will both be in school full day eliminating the additional expense, assuming you will access public education.
You may consider taking classes and paying for a babysitter, which will likely cost you $10-$15/hour. Without an income right now, you should focus your resources on your immediate needs and pursue your education when you can do so without adding to your financial burden.