DNS changes take a while to propagate and can cause some weirdness.
"ISP provider" can mean a lot of things though. You talking about your web host? Did you simply specify a new set of authoritative DNS servers at netsol?
When you go to ping, your local computer is going to do a DNS lookup. If you've looked yourself up recently, it's generally going to pull it from a cached value versus going all the way out to get the new value. How long that cached value will be honored depends on the domain's TTL setting. This may not have anything to do with the TTL of your ICMP error message, however--just a confusing coincidence.
Anyhow, I imagine if you flush your DNS resolver cache you might get your new IP when you ping
ipconfig /flushdns ping my.domain.name
And depending on whether your _local_ ISP's DNS server is a caching server or not, you may get the old or new IP....
Could also be other things that are a function of the new ISP and less to do with the DNS change.