If material is of suffecient depth and there are ample dranage holes at bottom you will not have too much moisture. Plants require several nutrients and elements in addition to N-P-K, might be time for a soil analysis. Unless compost is from diverse sources and well aged plants don't do their best in pure compost. Those plants are small enough to transplant so I'd mix 25%-40% native soil with your compost and put a few plants in that for comparison.

I see the wood fragments but i don't think they are sapping enough N at this stage to worry about sifting them out.

Let me show you what the compost looks like. My definition of unfinished wood chips may be incorrect (which I hope is the case!). First picture is the overall compost, second is what I sift out of it. It almost has the consistency of mulch. You think these are ok to keep mixed into the soil?

After seeing % of wood in mix,it need's at least half the wood sifted out. Return wood to composter mixed with green grass clippings or other green material to further breakdown.