When refinancing, don't take the first offer that comes around. Instead you should actively compare both interest rates and lenders.
If your property value reduces and you refinance up to 80% of the reappraised value, your original mortgage amount may be higher than this amount. Thus, the new loan will not be sufficient enough to help you pay down the existing one.
Many companies in this industry will do what ever they can to get away with charging you as much as they possibly can. Some of the ways they do this is not disclosing all the third party fees involved in a loan such as title insurance, appraisals, pre-paid tax and insurance escrows etc. It is important that you ask the loan officer you're speaking with about third party fees. If you don't they may not tell you and give you a good faith estimate that sounds fair, but at closing you'll find out that you have to pay a couple of thousand dollars more in fees you were unaware of. A good loan officer at a reputable company should have no problem disclosing all fees that pertain to your loan and should also make sure you understand what the fees are for.
If your new mortgage rate seems too good to be true then it probably is. Check for hidden fees in your mortgage that will make up that suspicious difference.