Both
flickr and
photobucket offer EXIF at the original sizes uploaded only. So you mention you do not like using compression. But if you use
flickr and
photobucket and then choose the size that AP allows (up to 1024 pixels), if that is not the size you uploaded, then the EXIF will not be available.
EG, you upload a photo at 4000 pixels x 3000 pixels. If you share that full size, then your EXIF will show. but as soon as you select a different size to share (eg 2000 pixels, 1024 pixels, etc) the image will no longer include your EXIF.