Update, 7-24-17: in working with images on my other blog tonight I discovered Photobucket has write-protected ransomed images so that even if the method described below works for you, you can only view ransomed images in your browser. Once you download them, they're not viewable on your computer. They also cannot be viewed if you upload them elsewhere. As I say in the post following this ETA, download your albums. Images in downloaded albums were not write-protected - at least not "as of a few weeks ago" when I grabbed my albums.
As I said to Andrew Ducker, it's slow and awful, but until Photobucket disables the ability to use this trick, if you have a copy of Firefox to travel to the image(s) in question, it works even if you're not logged into Photobucket, but does not work on images with the word "album" in the URL, only on images with the word "gallery" in their web addresses.
Which I learned only after my profile meme got ransomed sometime this week, which I caught on to just minutes before posting this. Photobucket images on my blog might be gone now, too, but I don't have time to check.
For Photobucket URLs with the word "gallery" in them, only...
- Right-click the ransom badge in your post (for expediency's sake, I'm linking to Andrew's images for testing, so you can try the steps out yourself, as I had no ransom badges of my own when I began this)
- Right-click the ransom badge on the Photobucket page you're brought to
- The original image has now become visible
Right-click the image and click "Save as" to download(see ETAs at top of post and below for more info) Screencap away - lotsa gotchas, though, if you go this route
- Once you follow the above steps, hit your browser's back button. The original picture (not the ransom badge) will show up on its Photobucket page and the post it was added to.
These steps have only worked for me so far in Firefox latest (version 54.0.1) so they might not work in other or older browsers. They do work in both Firefox's regular and private browsing modes. Using Opera with adblock results in a shocking Photobucket directive to install an anti-Adblock script called BlockAdblock, aka FuckAdblock, and links you to a site that I won't share because Web of Trust gives it such a low rating. "Photobucket teams with FuckAdblock" needs to be my next post title, as that's kind of news in itself.
ETA, 7-26-17: Using Chromium (like the barebones version of Google Chrome, which I had to use tonight to deal with Wordpress's PHP-from-hell backend burning up Firefox) the above steps not only won't work, but clicking "Save As" will result in downloading Photobucket's actual ransom badge.
As for other ways to access Photobucket images, Fix Photobucket and the photobucket embed fix app are both dead. You can spam me in the comment section with any apps, tools or websites that actually still work, but I'm quite picky and will try to check a tool before I publish any comment on it. Comments initially will stay screened.
ETA, 7-26-17: killerweasel has found a script via Twitter which, if you own a Wordpress-based site, replaces the images for you. From their web page: "PhotobucketScraper is a script to download all images to your server in the original folder structure." For website owners, this or a similar plugin might be the best way to go.
Downloading Photobucket Albums
Edited on 7-24-17 to add: download your albums. See instructions at next link. See Update at top for more info.
I've had no problem downloading Photobucket libraries, but there are reports that a few people have been unable to. If you don't block ads that could be one reason why; the page basically never finishes loading and will load one ad after another (up to a dozen) which locks your browser up and slows everything down while you're trying to access images, download libraries, or even copy and share file links.
Some people claim Photobucket's mobile site is much faster. There's also a Photobucket app that's said to load up even quicker than Photobucket does on the Web.
As for moving Photobucket albums to a new image host, you might want to skip moving them to imgur. Imgur's ToS prohibits creating libraries that have already been linked to from other websites and will resize images on direct link, making them basically useless.
You might not want to bother with TinyPic, either, because it's already owned by Photobucket.
So I can tell you what others have suggested and what I'm doing: others mainly suggest getting your own domain name and a hosting plan to host images yourself. From what I gather, self-hosted image libraries (not some roll-your-own solution like adding them to posts, or buying a Wordpress-hosted plan) involve buying or at least adding a separate tool or utility to the backend of your website.
Other ideas include using a free Blogger account, then creating individual posts (Blogger allows up to 10 images per post). Kind of awkward...then there's free Wordpress, which along with Dreamwidth's hosting, is what I use for some of my images. Free Wordpress has 3GB of free storage space, and individual posts are automatically created for each image. These posts aren't displayed publicly nor are they indexed by search engines, and you don't need to know their addresses since you can find all your uploaded images in the Media Library.