We consider the communication of time-sensitive information in NextG spectrum sharing where a deep learning-based classifier is used to identify transmission attempts. While the transmitter seeks for opportunities to use the spectrum without causing interference to an incumbent user, an adversary uses another deep learning classifier to detect and jam the signals, subject to an average power budget. We consider timeliness objectives of NextG communications and study the Age of Information (AoI) under different scenarios of spectrum sharing and jamming, analyzing the effect of transmit control, transmit probability, and channel utilization subject to wireless channel and jamming effects. The resulting signal-to-noise-plus-interference (SINR) determines the success of spectrum sharing, but also affects the accuracy of the adversary's detection, making it more likely for the jammer to successfully identify and jam the communication. Our results illustrate the benefits of spectrum sharing for anti-jamming by exemplifying how a limited-power adversary is motivated to decrease its jamming power as the channel occupancy rises in NextG spectrum sharing with timeliness objectives.