Should I pay out of my own pocket to maximize my settlement

Should I pay out of my own pocket to maximize my settlement offer or should I just let the third party insurance cover my medical bills. I was just in an auto accident last week. I have had one ER visit and a few chiro visits. I have sprained my neck. My car was totaled but it is an older car. Damages were well over $8000 to the car. There is no dispute that this was not my fault but I don't want to be at a disadvantage if I let the third party pay. I think it's not too late to have the ER and the chiro not send the claim to the third party. Please let me know if you think it matters if I pay or if the third party pays regarding my settlement offer. I'm in Texas.