Adobe Raw update

If you look here:  Cameras supported by Camera Raw

You'll see the D5600 was first supported with ACR 9.8  (ACR = Adobe Camera Raw, the PS camera raw plug-in)

If you look here:  Camera Raw plug-in and Adobe application compatibility (Note 5, specifically)

You'll see that ACR support for CS6 was discontinued after ACR 9.1.1.

CS6 is many versions obsolete by now, so Adobe finally stopped spending technical and marketing resources on it as CC has long since been the current series.

If your operating system is not too old that the program won't run on it, the one non-pay-for choice is to use the DNG Converter 9.8 and newer to create DNGs from the NEFs and those DNGs but not the NEFs will open in the older CS6 ACR plug-in version.

The DNG Converter is available, here:  Adobe software and product updates

Otherwise, the pay-for option of subscribing to the PSCC + LR CC for $120/year (billed $10/month) in the US, will get you the latest and greatest Photoshop and Lightroom versions.  Before actually paying anything, though, you'll probably want to do the 1 week trial of CC to make sure it is compatible with your OS and computer hardware, if those are both older.

To do the 1-week trial you can install Creative Cloud from there and just install PS CC (for now):

Lightroom and Photoshop | Adobe Creative Cloud Photography plan

Personally I like Lightroom better than using PS with Bridge and ACR, at least for the bulk of my photos, then I use PS for finishing touches.