How to recover data from Time Machine

I have a late 2011 iMac with an internal drive running 10.6. I also have an external drive that holds my Time Machine backup data plus a small partition with OS 10.7. For the last 3-4 years I have been running the iMac off an external Thunderbolt drive running 10.13. Recently that drive crashed totally. I recently installed another external drive in place of the dead one, put OS 10.11 on it and moved my data and applications to the new external drive from the backup drive. What got moved, however, was not the recent data and apps but data and apps dating from 2014 on the old internal drive when I moved daily work to the external 10.13 drive. Strangely, if I open the old internal drive showing on my 10.11, that drive shows the former current app list and moreover I am able to open those more recent apps, no of which showed up in the migration. However, in every case each app can be opened as a first time opening and asks for the original password. All those passwords I had placed in DevonThink, but now when I open that app, it is empty. My question is how can I retrieve apps and files that were omitted in the recovery that date from 2015-2018 and vanished in that dead drive. The data are in the Time Machine backup. There should be some way to access them. Thanks for any ideas or suggestions.