JamesSwanson

I use IrfanView
https://www.irfanview.com/
Can handle 16bit Scans. And Batch-Scanning and it is free. 
After that Affinity Photo
And i do my scan in tif.
 
Addition:
You can Scan in 16bit/48bit 
IrfanView can handle this
But only save in 8/24 bit

Affinity Photo should do what you want. The learning curve is steep if you have not done photo editing before. It's among the most complex things anyone ever does on a computer, so don't be easily discouraged.
Obviously Affinity Photo's competitors will also do the job. They might even be easier to use at first glance. Some even have automated features (that I've found don't work satisfactorily despite advertising hype).
However, if you strive to be a serious craftsman then Affinity Photo's emphasis on non-destructive editing is an enormous help in this kind of work. You can repair physical damage on the photo without making physical changes to the pixels of the scan. If you make an error, nothing is lost except time and nothing is ruined.
See my simple example at
https://forum.affinity.serif.com/index.php?/topic/77763-recovering-memories-gifting-friends/
If you have not done much scanning, read Wayne Fulton's classic work "A Few Scanning Tips"
https://www.scantips.com
including this page about old photos
https://www.scantips.com/restore.html
I also use VueScan for scanning photos as it gives more control than the software that came with my scanner. It includes some photo restoration functions that are best applied during scanning rather than later in an editor. If you are not satisfied with your scans, then you might give it a try.
https://www.hamrick.com
Again, the learning curve can be steep. Restoration is a combination of technical and artistic skills informed by experience. 
 
 
 

Just a couple of cents worth of advice:
Old pictures need care when scanning. The light from the scanner can cause fading so choose a real boring photo (one you won't worry about potentially losing) to do your testing with. Once you have the right idea about how to scan you can then go through all the old photos and archive them.

Hello @JamesSwanson and welcome to the forums.
I use VueScan for scanning my old photos. It can save the images as 16-bit tiffs. I would recommend scanning in colour, then using Afinity Photo for post-processing and retouching.
Photo can convert your image to black-and-white (greyscale). It can also enhance the contrast. It is better to leave any sharpening or clarity adjustments to near the end.
Use the dust and scratch removal tool for small imperfections.
Use the clone tool or the inpainting brush for larger imperfections.
I would suggest you try it out and, if you run into trouble, post your image here for us to see.
John

Hello James,

Affinity Photo is good for photo editing. 

Best greetings 

Anke