If I was arrested for writing a bounced check, can I go to the person who signed the warrant ask them to drop the case against me since I paid the money for the check?

Just to clarify, who exactly are you talking about? The person who signed the warrant? The person who filed the complaint? Or the complaining witness (the person you wrote the check to)? The person who signs the warrant is usually the judge. The person who signs the affidavit requesting the warrant is usually a police officer. The judge will certainly drop the charges if he finds there is no evidence to support them, but you will have to make your argument in court. The officer doesn't have authority to drop charges once they are filed. Once the charges are filed the prosecutor decides whether to prosecute or not. 

If you are talking about contacting the complaining witness, DO NOT contact him/her. I cannot stress that enough. My colleague from Texas knows the law, but I believe he understates the gravity of contacting a witness and encouraging them to drop charges. It won't be misinterpreted as witness tampering. It IS witness tampering. Naturally I recommend you hire an attorney to handle your negotiations. If you want to represent yourself you need to talk to the prosecutor. Understand, though, that making restitution is not a defense. 

You can go and ask the person to drop the charges, but the better practice is to have a defense attorney do it for you.  If the attorney does it for you, it is usually considered "plea negotiations."  If you go and make the request, it could be misinterpreted as witness tampering-- which could get you into more trouble.  This seems like a very simple issue to resolve.  You should be able to find a criminal defense attorney to help you for a relatively low fee.