Increased taxes, looking for refinancing advice

[Sarah B Beggs]

I am looking for some advice, I am a newbie and don't have any experience or knowledge in this area other then what I've learned over the past year.  This is regarding my first home purchase and I feel like this would be the best group to get feedback from.

My husband and I purchased our primary residence in March of 2020 with a conventional loan at 3.75. Our taxes are paid through our escrow account and were calculated at closing to be $1187.06. When we received our escrow analysis at the end of the 2020 year they were in fact $4797.19, leaving us with a total shortage of $3609.94 ($300 a month for the next year).  In addition to this, we also now have higher taxes due to an increased property appraisal. These changes now make our our payment  an additional $600 a month (went from $1479 to $2017). Would it be appropriate to try and refinance to help lower the monthly payments, or should we pay off the additional taxes and then try to refinance? We are building our emergency fund and the surplus of $3609.94 could also be used/kept there. This is a big increase for us, and not something we were prepared for. I appreciate all of your feedback and advice. Thank you!