It might be a little early still. Here in Missouri I will be planting my fall carrots the last weekend in July. Our average FFD is about the same as yours. Carrots are way sweeter when harvested in cool/cold weather. When grown in the ground they can be left in the ground and harvested as needed provided that the ground doesn't freeze solid. If well mulched the freezing can be mitigated well into December or even January in a mild winter here.

Sowing carrots in the hot weather is a little tricky. The soil needs to stay moist while they germinate and when they are tiny seedlings. They also germinate much much faster in the warm soil...like 5-6 days. What I do in my raised beds is moisten the soil well and then cover with cardboard. The next morning I remove the cardboard and sow the seeds and barely cover them with some fine moist soil. Then I replace the cardboard and start checking them for germination after 3-4 days. As soon as I see any germination I remove the cardboard and start watering daily with a fine mist or soaker hose.

It looks like we're in a similar zone. I also planted carrots in early April but harvested them all by late June. Waiting any longer makes them woody and earthy tasting due to the heat of summer. For fall carrots which are much sweeter, I'll plant that seed around Aug 20th and start to harvest in early Nov thru Feb. I'm no fan of container planting as the ground will keep the fall carrots in cold storage whereas the potting soil could freeze solid before Xmas.

My soil isn't particularly clay, but has a lot of small rocks, so maybe a raised bed?