With this proposal, we aim to continue P1032, the rapid measurements of relativistic parameters of southern binary pulsar systems using data from the Parkes Ultra Wideband Low (UWL) receiver. As the MeerTime project at the MeerKAT radio telescope ends in early 2024, observations with the UWL receiver will now form the backbone for extending the timing baselines of many pulsars that have been part of this project. Under these circumstances we have revamped the source selection in this proposal such that the UWL receiver is the most (versatile and) suited instrument to achieve the proposed scientific goals for the chosen pulsars. We will use the UWL data to obtain significant orbital coverage and timing baselines on these pulsars and obtain important constraints on orbital and temporal DM variations that will help in identifying the nature of the companion stars. Increasing the number of measured Neutron Star (NS) masses, as well as improving on current constraints has profound implications for NS internal structure and for stellar and binary evolution physics. We request pre-graded status for this project for the next semester.