Cellular Networks (CNs) are of paramount importance to enable Public Safety (PS) broadband services. During a disaster, due to damage to the infrastructures such as Base Stations (BSs), the access of PS Users (PSUs) to the core network can be disrupted even with the use of a relay station. To solve this problem and ensure the continuity of PS services under any circumstances, Proximity-based Services (ProSe) is a key enabling technology that allows Device-to-Device (D2D) communications to take place in licensed and unlicensed spectrum. In this paper, we propose a new architecture for PS Networks (PSNs), by using Non-Orthogonal Multiple Access (NOMA) technique and by placing the ProSe function and the ProSe application server in Multi-access Edge Computing (MEC). In this way, network availability for the maximum possible number of PSUs can be guaranteed. Moreover, they can access the required information with very low latency in the licensed spectrum. As well as they continue to operate in the unlicensed spectrum in a managed and secure manner. After the evaluation of the proposed architecture using the Simu5G network simulator, results reveal that by using NOMA, the spectral efficiency can be improved up to 28.8%. The benefits of using the MEC system in reducing latency and increasing reliability were clearly observed. In addition, we investigate the effect of increasing the number of requested applications and numerology on the task response time, which is a very important metric for PSUs.