Buying mudguard and pannier together vs separately
If your bicycle and riding style will permit it, I would certainly recommend getting "proper" fenders / mudguards which are close to the wheel and cover it a good portion of the way round.  These will do the best job at keeping spray away from you and from the rest of the bike.

You say old standard frame - so that sounds like it may well have the right attachments for fenders / mudguards.

I spent many years commuting year round in London - 12 miles each way, and spent a great deal of time and bitter experience optimizing my set up.  I tried different bike set ups and a variety of fenders / mudguards and by far the most effective were the SKS thermoplastic ones which sit very close to the wheel.  They stop the most spray going up your back / butt and over the bike.

If the weather is reasonably inclement the difference between ok and good fenders is a lot…in my experience of riding through driving rain, sleet and snow!