Henry - Yes, you could replace the current HD with a SSD but you don't want (can't fit) either of the adapters you have links to above, and frankly you really don't need them.
Depending on the age of your MBA you could have either a ZIF or Pin header connector'ed HD/Ribbon cable. Until you open up the unit you won't know which you have for sure (If you have never had your system serviced you can use the model numbers but I have seen people switch things so its best to check). Once you know you can find the correct SSD unit to replace your current HD. Make sure the SSD you order is the same size as your current HD.
Don't forget you'll need an external HD case to hold your new SSD to connect to a working Mac to prep it. I would recommend using a Firewire case as it will make quick work of the needed tasks (and here you might need the ZIF to Pin header adapter depending on what your MBA you have).
Last issue is cost of a SSD as big will be costly! Vs going with a replacement HD (even one thats larger). You may want to re-think going with SSD.