Should I move my educational loan from HDFC bank to any other bank that offers lower interest rates?

I took a education loan of 5 Lakhs in 2021 from HDFC Credila at 11.25% with effective tenure of 156 months, over the years they have increased the interest rate and as of today, it is currently at 13.7% with effective tenure of 178 months. So should I move my loan to any other bank that offers a lower interest rate? My confusion is because while the interest rate has increased, my monthly EMI outgo has remained the same . My monthly EMI was about 6,113 and over that I have made an additional payment of whatever money I can save and currently the EMI is about 5,785 per month as of today. The current outstanding amount is 4,39,126/-. Does it make sense to move it to another bank if I can make an additional payment of about 5,000 every month over the EMI amount? Will I be saving money if I do and is it worth the hassle of going through all the documentation process etc.?