There are definitely seeds and your tilling likely exposed more, as well as some weeds that may grow even more numerous after having roots cut up. However, if you are vigilant about keeping that are in check, then that should be OK.

You want your soil to be a bit crumbly and it may have large pieces from being dug up but you can also make them smaller by crushing them or using your shovel/hoe, so that when you dig it and move it around, it falls easily into place. Moist is not bad if it can do this, but wet or muddy is definitely not a good condition for digging. You might get away with a plant here and there or a seed here and there (eg I think onion bulbs would be OK if you dig up mud and chunk it back over the bulb, but that's just me), but you should not plan on the bulk of your day's work to be when there is wet ground or mud.

If it stays in large chunks when you dig at it, then it could stand to have some soil amendments to help it. Well that's kind of a misdirected statement as many soils can always use more amendments anyway.

If you are staying with gardening in that ground, then just go ahead and start planting when you are done with your soil amending. You won't ever totally get rid of weeds when growing in ground, so just start the garden. Over time, with your growing and weeding, you'll get it more and more under control.

Don't water till you plant things. First till in some compost, smooth out the soil, pick off weeds and grass from top 6", plant, watch for weeds and pull as you see them. If you plant in rows, you will know what is a weed. After a while, apply mulch and that will help suppress further weeds.

If you really want to know what to amend with, bring a jar of soil to your local cooperative extension for soil test and go from there.

Read a lot about gardening and have fun with it.