Is a written agreement between two parties and signed by both parties a legal binding contract in the state of Alabama?

That depends.  A contract is just an agreement, and most contracts don't even have to be in writing to be enforceable.  However, there are too many variable to be able to answer your question definitively.  For example, were both contracting parties of age and competent?  Was consideration (quid pro quo) exchanged?  Is there a claim of fraud, mistake, or duress?  Is the object of the contract legal (for example, gambling and prostitution contracts are unenforceable in most US jurisdictions)?