I have the same type of situation with my compost. It is ok to use the compost at any stage. So yes you can do either but I prefer to put the compost in the hole...

Sounds like for the most part, you're doing things OK.
I would add whatever browns (leaves twigs, shredded paper/newspaper etc) to the mix for composting.
The stuff you have already can both be tilled in and/or incorporated into the planting.
Happy Gardening! Nancy

Thanks, nancyjane and mckirk. I dug out most of what was in the bin, leaving the soggy part in the bottom and mixed a fair amount of dry browns into that, then replaced what had been on top with some more dry browns added in.

I'm eventually going to build a bigger wooden bin, but don't have the time to do so right now.

One thing you can do when you have a bin with a small footprint like that is to elevate it a bit off the ground. I had to do that with some food waste bins at the office because the coffee grounds and food waste just got too soggy. We put some bricks and blocks down, then chicken wire or hardware cloth on top of that, then the bin. The bins drained a bit better and did not get as soggy.

But you'll probably never get it perfect from top to bottom...if one end is dry enough the other will be too dry! Luckily compost is forgiving.

Update: On Monday I had a ton of kitchen scraps, including about 7 small tangerines and 3 large oranges that had all gone moldy. I put everything in the food processor (coffee grounds, onion skins, banana peels, corn husks, egg shells, cabbage leaves, carrot tops) and then mixed that slurry with torn up egg carton and dry crunched up leaves and miscellaneous flower bed rakings. That went on top of the pile after I turned it, and then I added a few more handfuls of crunched up leaves and rakings on top.

I went to turn it a bit yesterday afternoon, and for the first time ever, it was hot! Not just warm, but hot enough to warm my hands over it! Now I just have to figure out how to do it again and again!

I feel successful! :)