Pilot reported ground proximity obstacle alert due to ATC improper clearance phraseology resulted in evasive action to avoid a radio tower near CGI airport.

Narrative

We were on the descent in to the destination airport. We were given the local Center ATC frequency which controlled the airspace to provide clearance for an approach at our destination airport. I was the PM (Pilot Monitoring). I checked in with ATC who gave us a descent to 3;000 feet. We were approximately 30 miles to the North of the airport. ATC was heard talking to multiple aircraft at a field nearby and seemed task saturated. ATC said in a non standard; not instructive radio call telling us to expect Runway 28 at the destination and that if we wanted to we could turn to XXX heading (left heading) to set up for the approach in to the airport. The way he worded it wasn't a 'Call sign; turn left heading XXX for vectors for Runway 28'. It was very non standard and caught me off guard. Due to calm winds at airport; I ignored his suggestion for a left heading and asked for the opposite end of the runway; Runway 10 which we had already set up and briefed. ATC replied back saying unable due to traffic in the area. PF (Pilot Flying) and I then began our duties of changing our runway and briefing the differences. While this was happening we were still direct to the airport. As we were leveling off at 3;000 feet we get the 'Ground Prox' flashing and the 'OBSTACLE' audio warning. I look up and see a very tall radio tower at our 12 o' clock. I immediately tell the PF to turn left heading as I informed ATC that we are in a left turn to avoid a radio tower. He doesn't seem very concerned about the situation and says the current heading we are on will work for us to get set up for the visual approach to Runway 28. Unfortunately due to hazy conditions; we were unable to locate the field for a visual approach. ATC appeared to be stressing to us to locate the field; offering the location of it several times. We assumed this was to get us off of his mind so they could focus on the several other aircraft at the other airfield nearby. I briefed with the PF that the safest thing we can do right now is have him vector us for the GPS approach for a safe approach to our destination. PF agreed and we told ATC we are requesting vectors for the GPS approach to [Runway] 28. We landed safely at our destination minutes later. On the ground we debriefed the situation and looked up the radio towers on the VFR chart of our app; showing the towers around 2487 feet; with a minimum altitude in the area of 3600 Ft. It appears we were given a lower altitude in the airspace that brought us within 500 feet of the tower which triggered the Obstacle warning.Improper ATC communication; task saturated environment for both ATC and flight crew (lack of visual scanning due to setting up for last minute runway change); lack of proper arrival briefing of obstacles outside of the terminal area and knowing lowest safe altitude we could descend to for the area we were in. I think this could have been avoided by having proper ATC instructions instead of a laid back suggestive discussion. Besides that; looking more in depth at our arrival corridor for obstacles as part of the brief would have assisted us in knowing what to expect in the area.