Answer: Damping off is actually more dramatic, it attacks seedlings when they are very small and they all keel over suddenly, rotted off at the base by the soil line. They can't be saved once this has happened. In this case, though, it sounds like a fuzzy white mold sometimes seen on the soil in pots of seedlings that have been kept overly moist.
The peat pots will eventually disintegrate if kept moist and mold growth can be a natural part of that process. If the mold is only on the pot itself, I would not worry about it. This is especially true since you are going to be planting the plants in the garden soon.
If the mold is on the soil surface, gently scrape it off and cut back on your watering a bit to try to allow the soil surface to dry out between waterings so it is not staying soggy for long periods of time.
Increasing the air circulation will also help, so you might spread the pots out a little bit so air can flow between them. Setting the plants outside while they harden will also help improve the air circulation. Some gardeners find that a gentle watering with (cold) chamomile tea helps dispel this type of mold and can also stop damping off.
If the mold has engulfed the plants, gently clean it away and discard any plants that may have deteriorated as a result. As soon as the plants are hardened, plant them in the ground, removing at least the top third of the peat pots, more if you can without tearing away roots.
The basil will need special potection if there is frost and/or on cold nights, but the others will tolerate a good bit of cold once they have been conditioned.
If you transplant with care, and the plants are neither undersized nor rootbound when you do it, there is little need to worry about undue root disturbance which I think is the main reason people like the peat pots. Unfortunately, peat pots can be difficult to work with, in my experience they tend to become water logged and encourage mold but then will also tend to dry out and wick water out of the soil mix.
Good luck with the mold and the seedlings.
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