I've seen renew contracts getting cheaper, not the other way. Moore's law at work. If your provider is asking for more, change the provider.Copy all content from one site to the other, put an entry on /etc/hosts for your domain, pointing to the new IP, and test everything.
Before the move, change the TTL for your domains for a small interval (around 5 minutes). One of two days later, when the DNS changes had propagated and your tests passed, change the DNS records pointing to the new IP and restore the TTL to the old value.Why change the TTL twice? The first time is to tell every DNS server, cache and client that a change is imminent. When you do the change, very few clients will have the old IP on cache, as the record expires fast. The second change is to bring things back to normal.