Your current insurance company will definitely not be contacted. The only time your company would be contacted is when you call to cancel your policy. Make sure that you do not cancel your current policy until your new one is active to avoid a lapse in coverage. You might also consider contacting your current company to see if there is any way you can save money on your policy. Shopping around for insurance is important because rates can change often and you want to make sure you are getting the biggest bang for your buck.
When it comes to your credit score and insurance quotes, your credit will not be affected. When carriers run credit, it is usually not until the end of the purchasing process that an exact rating is not obtained which is why your score will not be impacted. What it can impact is your rate. Most carriers will rate on credit but there are a few that do not if your credit is not the best.
Also, take a look at your Insurability Score for more information on how your credit will impact your rates as well as other rating factors for insurance.