First, there is a difference between an assignment and an obligation to assign. If your "assignment" document states "agree to assign" or similar language, then it is merely a promise to assign rights in the future and "not an immediate transfer of expectant interests" (Board of Trustees v. Roche, 583 F.3d 832, 841-42 (Fed. Cir. 2009). This means that a subsequent and proper assignment needs to be executed in the future. To effect an immediate transfer, your assignment should state "do hereby assign" or similar language (cite).
Second, patent assignments should be recorded with the US Patent & Trademark Office within three months of execution. 35 U.S.C. § 261. If you fail to record within three months, you should still record the assignment because late recording is better than never recording. The purpose of recording is to provide notice to the public and to prevent a bona fide purchaser from creating a potentially costly mess over the ownership of the patent.
Third, you can record patent assignments for free on the USPTO website (cite). If you wish to submit paper documents, it will cost $40.