You are running into a couple of different issues here.
First, an attorney is retained as an advocate; and the attorney is ethically bound to act in what the attorney feels is the best interests of the client that retained them. However, if both parties want to retain an attorney to review a contract for them, an attorney could ask for a conflict of interest waiver. Some attorneys will agree to do this; others will not.
Second, the cost of reviewing a contract can vary widely, depending on (1) the length of the contract; (2) the technical nature of the contract; (3) the state law where the contract is being entered into; (4) any federal questions that are in the contract; (5) insurance and indemnity obligations; (6) and regulatory issues in the contract. Since you are dealing with financial software for the health care industry, you can probably expect regulatory issues to be a big consideration.
That being said, you should be able to find an attorney in your area that would be willing to review the contract you drafted for a flat (fixed) fee. What the attorney would charge would depend on where you are at; and the attorney's experience level. Make sure that you get the attorney's agreement in writing, and make sure that if there are going to be any extras (for example, if the attorney suggests changes) that the cost of drafting and/or a second (or more) review are set forth in your agreement with the attorney.
I hope that this gives you some guidance on the type of attorney to find to do this important work for you.