Do I first need to upgrade from 6.11 to 6.2x on the existing server?
I would recommend that you do it this way
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Then can I install a fresh copy of DP 8.0 on the new server and then import the IDB database from the old server to the new server?
No... DP 8 uses an entirely new IDB, so it is not so easy to just pick it up and move it as was in older versions
DP 6.2 gives you a direct upgrade path to DP 8.0 that an upgrade from 6.11 does not offer
At this point, where DP 8 has so many problems with their IDB, I recommend that you upgrade to DP 7.0 on your new server, you should be able to just move the IDB from one server to another, and run
omnidbutil -change_cell_name
There is a new patch bundle for DP 8.11 which fixes many of these problems, but, I would wait to see if it introduces new ones