I think it's safe to say carrots are a challenging veggie to grow. There's lots of good, and interesting advice. Just like most gardening questions the answer is it depends. your climate, your soil, your seeds. time of year...
The good news is seeds and pretty cheap, and carrots will grow in containers, or pots, as long as they are deep enough for the variety you choose. Maybe start a pot several different times of year. Use a good potting soil, so it's nice and fluffy, holds water without getting too wet. Then try some of the different methods. I'm in zone 9B, so what works for me may not work for you.
For me there was a game changer, and a bonus. The game changer was the board trick. I went from getting little to no germination to decent germination. Then I discovered vermiculite. Now I make a divot where I want my carrots, wet it well. Sprinkle a thin layer of vermiculite wet it. Then I sprinkle the carrots seeds. I try to space them out the best I can because I hate waisting seedlings (I still have to thin them) Then I sprinkle a thin layer of vermiculite on top Mist it so the vermiculite gets wet, but doesn't displace the seeds. Now I place a board on top. I've tried cardboard presoaked and staked down. This was better than not covering them at all, but not as good as the piece of wood. I wounder if the weight helps? Now I don't mess with it for a week. If it's hot and dry I may mist under the board, and/or water around it. This is not normally necessary because I don't usually start carrots when it's hot. Once I see several little seedlings I will remove the board. It's important to keep them well watered to encourage more to germinate. For me once they look like carrot tops I no longer have any problems. I will wait until they are 2" or 3" then thin them so they have enough room.
Good luck and keep trying, sooner or later you are bound to find what works for you.