We have great success installing the old or suspect drive in a USB external enclosure... which you can purchase for $7.00 to $35 in the USA, depending on quality and size... but you need one with its own power adapter to run a suspect drive successfully...
Then, since you already know what is on the drive, you can plug the external drive into your USB slot... the working computer will identify the external drive by giving it a letter, then drag and drop all the files, a few at a time... onto folders you have already named in the new system.
Do NOT turn off the external drive until done, for best results... because you don't know how much longer the old external drive will last...
then burn to a DVD all the files and photos you have copied over. Inspect them to assure they are good and working files.
Now you can do whatever you want with that old drive, but your files are safe... as you nnow have a copy on a hard drive, and have burned copies to DVD's.