Ask HN: Financial advice) Managing student loan consolidation and auto loan

I am a recent college graduate, having completed my degree a year ago. I currently hold $43,000 in private student loans, with varying APRs: $11,000 at 11.5%, $22,000 at 13.5%, and $8,000 at 7.75%. I am seeking advice on consolidating these loans to secure a better interest rate. I have explored options with Earnest and received an estimated rate of 6% over five years. Notably, their application indicates no prepayment fees, which aligns with my goal of paying more than the minimum monthly payment. Specifically, I intend to allocate $1,200 per month towards these payments, exceeding the minimum requirement of $880, to expedite repayment and minimize total costs.

Additionally, I am planning to buy a car which will require financing through some auto loan. The car I am considering is priced at $23,000, and I am contemplating a down payment of $5,000 to $10,000.

I seek guidance on the optimal sequence of actions regarding these financial decisions. Should I prioritize consolidating my student loans before financing the car, or vice versa? I am concerned about the impact on my credit score, as both actions will affect it. Is it advisable to do both within the same month so that the previous loan doesn't get reported on my credit report?

For context, my current credit score is approximately 733.

Thanks!