Is it legal to issue partial distributions of $4,000 to each of us using the money in the checking account?

In California, you may make trust distributions at any time unless the trust restricts when they can be made. Apart from that, the distributions must be fair to all beneficiaries, which is usually satisfied if each beneficiary is receiving an amount proportionate to their beneficial interest; e.g., a 50% residual beneficiary should get half of the amount to be distributed. With a large amount coming in from the sale, you should be safe distributing as proposed.