Third party liability settlement not covering the full cost - Budget

Last month I got involved in a accident at an intersection in which I was at fault after failing to notice a vehicle coming from my right. The third party lodged a claim against my insurance, which was totally correct. After 3 weeks budget legal department has been pushing back offering a settle of less than R6k while the VW assessment center gave a quote of more than R14k. The third party has therefore been contacting me demanding a full payment. After contacting budget legal department, they clearly told me they are only going to pay the amount stated. Isn't third party liability cober supposed to cover the full damages to the third party? Why is budget now only willing to pay so much less than the damages? The third party is getting furious with me now.