So in all such situations it is always advisable to replace your web hosting service provider with a new one. The first thing that you need to do is to take a backup of all the data that is hosted on your current web hosting server. You can do this by using a tool called Filezilla, wherein you need to enter the FTP details and then you can download all the files to a folder on your hard disk.
The next step is to signup with the new web hosting service provider without terminating the services with your current web hosting service provider. The new web hosting service provider will provide you with all the details to upload the files to their new server. You can again use FileZilla here to upload the files from the folder on your hard disk to the new web hosting server.
Once this is done, you must check if the website and its webpages are opening properly. You can check this with the IP address provided by the new web hosting company. Simultaneously you can setup your email accounts as well. Once again if all is fine, then you can make changes to the DNS information which is held by the registrar from where you purchased your domain. The process for the DNS to change can take anywhere between 48 to 72 hours.
In case you need to setup your email accounts on the web hosting server then you might need to configure your email software so that it can contact those accounts. Do not delete your old email account, as it would take some time for the new email accounts to get propagated.
In a few days you will now be able to access your website through the new web hosting company using its domain name. Once you start receiving emails through the new server, and everything is fine then you can go ahead and terminate the services with your current web hosting company. In this was you will ensure that the down time of your website is minimized and also you will be in touch with your clients!