In my experience, trying to fix this from your current data, will take any time you want, and there's no warranty, that your result is identical to the initial data. It might be easier, to wait until the database is available again.
The basic problem is, that the conversion to a simple CSV format is not bijective - there's simply no direct mapping back from the CSV file to the original data. You can try to cut this problem down using some heuristics, but it will need a certain amount of scripting or programming. It also depends on the platform and tools you have available and - last not least - on your skills.
You might have some records, where no field value has an embedded comma. Extract these first, even if there few, it's a starting point.
May be you find a heuristic to distinguish at least some of the "embedded" commas by field separating commas.
You could try to identify data values of characteristic columns, e.g. mail address, phone number etc. Working from there, you could narrow down the number of columns which need more careful inspection.
Proceeding stepwise in such a manner should allow you to transform your data into a format that is better suited, e.g. using tabs as field separators.
You should also think about extracting only some columns and a unique key value (record number?) in the first step, and later matching the different pieces.
Write down, document, keep a record of what you are doing. Otherwise you will make things worse instead of better.