First off congratulations on the purchase! Yes at closing you did pay from 4/19=7/1 based on the originally assessed value of 853k. Yes you will get this back. Let’s work in reverse a little bit, that makes the explanation easier. As the market ran up, people bought homes that were more expensive then the previous purchase. So a home that was originally purchased at 600k then resold at 800k. At closing the buyer would pay the difference (like you did chance) but at the 600k assessment. Then a few weeks later the buyer would receive a friendly supplemental tax bill from the assessor that would cover the difference for that time period (in your case from 4/19 to 7/1) of the 600k to 800k assessed value. From then on all future tax is based on the 800k assessment.
Now to your case. It will work in the inverse manner. Several weeks from now you will receive a check (at least you should) from the county assessor. Your home will be assessed at the new value. You should not have to do anything. If you are very concerned you can call the county assessors office. I would wait a few weeks but if you want you can call them now.