Please do not yet replace or redo any of your work! I am having a look at the issue and it seems to be independent of 3.6 vs 3.7 changes.
It seems as if I can reproduce the problem and error message you are describing when using the 3.6 unitypackage spine-unity-3_6-2018-10-18, so it seems to not to be related to 3.7 upgrade but some other changes that seem to have affected both branches.
The error you described is due to clipData.animationReference being null, so in the Timeline Editor window when adding a clip to the SpineAnimationStateTrack this clip is set to null and then dereferenced - I will fix this bug, it should not be issued (but can be ignored rather safely). You could select the clip in the Timeline and then in the Inspector set the parameter Template-Animation Reference
to your desired asset.
Could you please describe what your previous workflow was, how you added the "SpineAnimationTrack" etc, and create a screenshot of the elements in your Timeline Editor window?
Could you please also tell me which version of the 3.6 unitypackage you have been using? This would help a lot as well!
I have created a ticked for the issue here: [unity] Unity Timeline Editor error · #1257