First of all, you haven’t provided enough information to know how to answer the question with any precision. For one thing, you haven’t said if you are suing in small claims court or in a regular court (municipal, superior, city, etc.). Different rules apply to small claims court proceedings. If you are in small claims court, you should have received some sort of instruction sheet when you filed advising what type of evidence can be presented at the hearing on the matter.
Assuming that you are not in small claims, the only way to guarantee that the recollection of your witnesses can be used to prove the existence of the contract is to have them present as witnesses when you try the matter.
There are various ways of trying to preserve testimony when you are worried that it won’t be available at trial. The most obvious one is to depose the witness. This involves having a court reporter take down stenographically and provide a transcript of the procedure of each side to the case asking whatever questions they wish (subject to some very loose evidentiary rules) and the deponent answering them. This is part of the “discovery” phase of the suit, usually done so that the parties can find out just what probable witnesses would say at trial. But when a witness is “unavailable” at trial (dead, hospitalized, far away, etc.), a deposition transcript can often be admitted into evidence.
There are some minor problems with doing this, usually related to the fact that the evidentiary rules about deposition questions are looser than the rules at trial. One of the examples of this might well apply in your case: you are wanting your witnesses to provide “hearsay” evidence, that is, evidence of a statement made out of court offered to prove the truth of the facts contained in the statement. You want the witnesses to say that the defendant agreed to a contract. You may or may not be able to get that evidence in at trial; the general rule is that “hearsay” evidence is not admissible, but there are a lot of exceptions, more than one of which may apply here (and I am NOT going to go into an analysis because it isn’t important). The point is that, while you certainly can ask the witnesses in the deposition about what the defendant said, their answers as recorded at the deposition may not be allowed to be admitted into evidence at the trial.
Several posters have suggested the use of an “affidavit.” This is a sworn statement by a person, declaring something to be true, under the penalty of prosecution for perjury if they are lying. The affidavit is written up, and they then proceed to sign it, usually in front of a notary public who then establishes that the person signing the affidavit is who they say they are.
Affidavits, however, are rarely admissible at trial. This is because the opposition has no chance to cross-examine your affiant (the person making the sworn statement). One of the main principles to our type of law is that every party in the process gets the chance to confront the people who are testifying against him/her. You can’t do that if all you have is a signed piece of paper and no affiant at the time of trial. Affidavits are usually used to support some assertion that has to be made for a preliminary action in the case, such as the issuance of a subpoena, or a court order of temporary nature.
Since we don’t know enough about your situation, I can’t say what you should do (and wouldn’t anyway, for the usual reasons (I’m not your attorney, you haven’t engaged me to give you specific legal advice, etc.)). I do, however, suggest that, if you are not in small claims court, you get ready to produce your witnesses at trial.
And no, as a general rule, you do not get to recover the cost of legal proceedings when you win a case. So your legal costs are not “tacked on” to the award, in most cases.