Basically you have several options, one option is to retain current server to allow the DTS packages continues to run from there, then proceed other databases upgrade to SQL2008. But you have to change all DTS packages connection strings to point to the new SQL2008 databases.
Second option is to migrate all DTS packages to the new SQL2008 environment, but run them as DTS packages. In this case, you have to setup all the related scheduled jobs manually, because the DTS functions are not the same like in SQL2000. During the course of migration of the packages, there are quite a number of things you will find not working anymore. It is because Microsoft only provides a backward compatibility channel for people to choose to continue to run DTS packages as per before, but there are so many things that not being taken care of. E.g. if your SQL2000 get used to use UDF using "System_Function_Schema", then it won't work anymore.
Third option is to convert or upgrade all DTS packages to SSIS packages. You can run the Migration Wizard which provided by Microsoft, but you will find it doesn't do the job completely. There are some tasks the wizard could not do, e.g. if your DTS packages using "ActiveX Scripts tasks", then the wizard only creates a blank task, all your scripts inside are useless. Or if your packages are using "Dynamics Properties Tasks" to pass values via global variables, then you will find they will not work anymore. All these problems will cost you a lot of time to amend the packages one by one.
Four option is to seek for some external software like DTSXchange, but they will charge you based on number of packages need to be upgraded. If your company can afford that, then that could be the best option. If not, then you have to find your way.