Argthtjtr

Here is the sentence:

The clock is ticking towards a new year even though we’re still
savoring the months that are now behind.

I am not sure how the lexical meaning "enjoy or appreciate" would fit here.

I have to take this on its own as you have not explained the context with the sentence before and after. Note my spelling used here is UK savour as in your title however question body uses US savoring

We would commonly see the phrase "towards a new year" either at the end part of December or any other countries end of calendar year.

There is an underlying hint that there are thus less favourable months ahead and in the northern hemisphere we can expect a potentially harder time weather wise.

Customarily this would be after Summer fruits and wine, Harvest Festival, Thanksgiving, Christmas, those months in which we have celebrated our bounty especially via taste.

e.g. to savour ones food at a time of enjoyable savouries.

Late in December we spend a lot of time on reflection of what the last 12 months have meant to us, as we prepare to make resolutions for the next.

e.g. we appreciate our achievements.

In summary it is a time for enjoyment and a time of appreciation thus both combined are all found in the one word "savouring"