The info that is stored in the ODBC manager is stored/accessable from within the registry. You could simply make your modification to the ODBC DSN there. This would automate your process and allow you to continue without a restart of the application.
Look in HKEY_Local_Machine\Software\ODBC\ODBC.INI
for your existing DSN's I'll attach below a small code snippet (that I borrowed from someone else) for creating/modifying an Oracle DSN. But in relity, you need only to modify the settings you find in the registry key.
var
Reg: TRegistry;
begin
Reg := TRegistry.Create;
try
try
with Reg do
begin
RootKey := HKEY_CURRENT_USER;
if KeyExists('Software\ODBC\ODBC.INI\NEWDB') then
//presumably it is correct
else if not CreateKey('Software\ODBC\ODBC.INI\NEWDB') then
ShowMessage('Error in Creating ODBC DSN
HKEY_CURRENT_USER\Software\ODBC\ODBC.INI\NEWDB')
else if OpenKey('Software\ODBC\ODBC.INI\NEWDB',False) then
begin
WriteString('Driver','C:\ORAWIN95\ODBC250\sqo32_73.dll');
WriteString('AsyncSupport','Disabled');
WriteString('Server','T

BSERVER:ORCL');
CloseKey;
if not OpenKey('Software\ODBC\ODBC.INI\ODBC Data Sources',True)
then
ShowMessage('Error in Creating ODBC DSN
Software\ODBC\ODBC.INI\ODBC Data Sources')
else
begin
WriteString('NEWDB','Oracle73 Ver 2.5');
CloseKey;
end;
end;
end;
except
on E: Exception do begin
SHowMessage(E.Message);
end;
end;
finally
Reg.Free;
end;
end;