Dallas Fire Rescue generally does send both an engine and a medic to a medical call. What generally happens is a call is dispatched over the DFD Dispatch channel (460.57500) using the Locution automated dispatch system (it's also sent to the appropriate MDC screens). The appropriate medic units will respond on the Rescue Response channel (453.90000) and the appropriate engine/truck/chief/etc. units will respond on the Fire Response (453.87500) channel. Once all responding units indicate that they're in route, the response channel will repeat the call (with any additional information that has been passed along) and announce that all units are in route. Once they arrive on scene, they may be told to move to the assigned simplex ground channel for that incident.
Please note that the ground channels are simplex and not heard unless you're close to the scene. There may be more than one incident working on a ground channel if sufficient incidents are happening at the same time. They space out the assigned incidents to help eliminate (or at least reduce) interference on that channel in situations where multiple incidents are sharing a channel.
They have their CAD (Computer Added Dispatch) on-line here ACTIVE INCIDENTS
which may help you find out what's going on nearby if you missed a dispatch. The list may be filtered for security, but I've seen many types of incidents listed so the filter isn't very restrictive if they do use one.