May Allah reward you for your pursuit of beneficial knowledge.
Islamic banks and financing institutions structure their home-purchase contracts in a different way than conventional mortgages. While the end result (in terms of the amount you pay) might not be much different, the Islamic contract differs in important respects, involving issues such as details of ownership, liability, responsibility, and clear stipulation of a single, known price. The following, brief paper may be helpful to you for an overview of what has come to be called “Shari`ah-compliant financing.”
Therefore, the short answer to your question is that it is definitely advisable if not required (wajib) for you to look into re-financing your existing, conventional mortgages through a Shari`ah-compliant alternative. However, the details of the contracts of each Muslim financing institution vary; some may be satisfactorily Shari`ah compliant, while others may have problematic or controversial details in their procedure. In general, it is advisable and an established sector-wide, international practice to look for an institution that has their products reviewed and approved by a Shari`ah Board, an expert body consisting of a minimum of three qualified and trustworthy Muslim scholars (fuqaha’).
And Allah knows best.