What are you growing? I've had plenty of luck transplanting seedlings from overcrowded cells, but I usually "pot up" - I start with what you've described, usually in late February/early March. Then, around the beginning of April, once my seedlings have their first set of true leaves (the second set of leaves, which will look like the leaves you recognize from this particular plant), I'll wet them down well and separate them into individual pots or cells, and let them keep growing until it's time to plant them out (here in Maine, that's the middle-end of May). I've had good luck doing this with nightshades (tomatoes, peppers, etc) and brassicas (cabbage, broccoli, kale, etc). I feel like tomatoes particularly need their own pot eventually to get nice and big and sturdy to get a start on the season. Cabbages and broccolis I've sometimes just separated out and planted into the ground with good results. Onions I plant all crowded, then just plant out - they do fine.
So I guess my answer is, just go for it! Give those transplants some
compost tea when you transplant, and they'll probably do just fine! I don't think they need the soil you planted them in to survive...