Don’t have specific experience with that NCH product, but I do with other software they offer, and I was NOT impressed.
Someone upthread mentioned Audacity, which is free shareware, and it is very very good.
To me, the question is not about paying someone else to do the work, but about what kind of quality reording you end up with.
I wouldn’t buy a new turntable untill you listen to your oldest albums and decide if they are still in good enough shape to re-record.
On the stereo, you need to look for a pair of connectors most likely labeled “line out”. They will probably look like this. The white is the left channel and the red is the right channel. Most likely your computer has a different kind of line in – like this.
Here’s where you’re going to run into your issue. Many computers’ line inputs are mono - only one channel, not two. The connector looks the same in both cases. You will have to look at your computer’s documentation to figure out which yours is.
If you computer has a mono line input, then you have a choice. You can still use your old turntable, get what’s called a y-connector cable to run both chennels together into one, or you can start buying more stuff to get two channels into your computer.
How about looking at your stereo and PC and letting us know how you want to proceed. If your stereo doesn’t have a line out (pretty unusual but not unheard of), you’ll need to buy a new turntable with a USB output, and if you do that it will also make irrelevant the question of whether your computer’s “line in” jack is stereo or mono.
If you have a mono “line in” on your PC but insist on stereo reording, the new turntable with USB is also your best bet.
If you have a stereo line in out on your player, and a stereo line in on your PC, you just need a cable to connect the two, you need to download Audacity, and you’re set to go.
Let us know how you want to proceed, and I can post some more info.