How Ping is still working even if server name has changed?

probably there is stil a dns record available for serverA or it is still in the local cache of your computer.
There is no WINS on another server in your network?
Another option could be if you have put serverA in the local hosts file.

at the command prompt type ipconfig /flushdns

WINS may not be configured on ServerB, but is it configured on ServerA?
Remember the NIC setting on your localhost/workstation will determine which WINS server to use.

Run an ipconfig /all on the workstation and verify the name of the WINS Server.

You can download a tool from the Microsoft site called NBLOOKUP.
It works similarly to NSLOOKUP but queries the WINS server database.

P.S. When pinging ServerA, do you use the FQDN?
Does it respond with the FQDN or only with the short name?
If shortname, then you're doing WINS name resolution, if FQDN then you're using DNS name resolution.

From my understanding and I may be wrong but re naming your server once it was set up is a big no no. atleast on server 2003, I was told that a smiple name change would cause so many underlying problems that it was best to wipe the server and start a new install using the name you desire to change it to. Could be wrong though. also have you ran CEICW since the name change?

The Host Name is ServerA in "IPCONIFG /all "but when I do :

nbtstat -R
nbtstat -c: it still  show ServerA

When I ping ServerA I just use shortname. By the way the server is on DHCP then is it possible the Wins is from the DHCP server?

I need to know what is returning in your query...not what version of the hostname you're using.
Post a screenshot of your ping, and I'll show you what I'm looking for.

DHCP doesn't offer any WINS name resolution.
run nbtstat -RR and then check your cache.

did you try rebooting this machine to see what happens with nothing cached?
Does this problem occur on any other workstation?
It could be that there is an entry in the hosts/lmhosts files for this server.
Check these files in %windir%\system32\drivers\etc.

if you have already done ipconfig /flushdns
login to all DNS server and check whether any one of the DNS contains old record - identify it and delete it manually.

did you remove old server from the domain before making any changes or disconnecting it from network.  If not - DNS servers might have old records and may cause such issue

For the reasons I mentionned it let us suppose Wins is activated in DHCP.