Sony A TO E mount adapters

I am considering changing from an A mount A700 camera to a Sony A7iii mirrorless camera. As I understand it I can still use my Sigma A mount lenses with the use of an adapter. My question is will the the camera work fully with this adapter and also which adapter do I use there seems t o be 5 different adapters.

I had the LA-EA4 adaptor on my Sony A7II. I had mostly old Minolta A mount lenses and they worked fine. However, it wasn't long before I bought new native lenses and sold all the A mount glass along with the adaptor. The newer lenses focused faster and were optically better. Also, having an adaptor is extra weight and length and if you put a heavy lens on it you increase the likelihood of putting stress on it or the mount.
I don't know about the characteristics of your specific lenses, but bear in mind the A7III is full-frame and the A700 is APS-C so if your lenses were designed specifically for a crop sensor body they may not work well and you'll get vignetting. As to which of the five adaptors, again it depends on your specific lenses. Some only have the full functionality on the EA4, while some will be OK on the EA3. See https://www.sony.co.uk/electronics/support/articles/00021281 The EA1 and EA2 are for APS-C sensors only so you can rule them out. The EA5 came out later and has superior autofocus (allegedly). I read also that the older adapters don't provide autofocus on the newer bodies. I don't know if the A7III counts as 'newer' but even if not, and you planned to get a newer body later and still use the adapter then you might need the EA5.
I don't know about the characteristics of your specific lenses, but bear in mind the A7III is full-frame and the A700 is APS-C so if your lenses were designed specifically for a crop sensor body they may not work well and you'll get vignetting. As to which of the five adaptors, again it depends on your specific lenses. Some only have the full functionality on the EA4, while some will be OK on the EA3. See https://www.sony.co.uk/electronics/support/articles/00021281 The EA1 and EA2 are for APS-C sensors only so you can rule them out. The EA5 came out later and has superior autofocus (allegedly). I read also that the older adapters don't provide autofocus on the newer bodies. I don't know if the A7III counts as 'newer' but even if not, and you planned to get a newer body later and still use the adapter then you might need the EA5.