It's hard to diagnose a problem like this without at least good photos of everything so the following involves some guesswork.
Is a stone with coarse/fine surfaces enough for what I am doing?
Impossible to say with certainty because of the variables. Unfortunately stones (diamond abrasive plates as well) graded as coarse, medium, fine etc. are by no means equal. One maker's 'fine' will be very similar to another's 'medium', something that has tripped up many of us buying sharpening products from different makers :-(
In the product photo you link to the honing guide is being used on abrasive paper on glass and I was thinking there was every reason to suppose they would go to a finer grit than the 'fine' side of many stones. Looking further down the Product Highlights underneath say they include a piece of 15 micron film, which is approximately equivalent to 1000 grit. Unless you know the grit ratings for your stones and the 'fine' is 1000 or finer that's likely part of the issue here (but see Note at bottom).
Just in general, a simple combination stone like this can be good enough as the core of a sharpening system... I should mention that many people with elaborate (and expensive) sharpening setups would disagree with this! But for planer knives how fine the 'fine' side is could be critical, although other issues including how you're sharpening may be as important or more important. Which leads us to...
What am I doing wrong?
Possibly nothing, but let's see. If you didn't begin by lapping the unbevelled face of the blade flat as per the instructions then do that first. As in all similar sharpening operations creating a uniformly flat surface along the edge on the other side of the blade is vital to creating a sharp edge.
Then, when sharpening the bevel, the PDF includes this:
Hint: Apply pressure on the push stroke only. This will prevent a thin wire edge from forming along the leading edge of the blade. Continue this action until a satisfactory finish has formed on the bevel.
If you haven't been using push strokes only when sharpening do that also.
Final point, check your stones for flatness. For this operation it's vital your stone (both sides) is completely flat. If you check with a straightedge and you find there's even a slight dish on either or both sides then flatten the stone before you try sharpening again.
Stones can be flattened in numerous ways, the cheapest for you now is to use a sheet of coarse wet-and-dry abrasive paper on a very flat substrate. A good surface for this in the typical home is a kitchen countertop, generally made of thick, stable material held reliably flat by how it is installed. But don't assume it's flat, check it first before use.
How much sharpness is good enough?
As a general principle no edge used in woodworking can be too sharp. Obviously there are complexities, but all other things being equal sharper is usually preferable to less-sharp (again, see Note bottom).
What do you do for testing the sharpness of your blade?
I've never tested a planer knife for sharpness but think of them as a sharp edge, not just a planer knife and test accordingly.
If they aren't good enough to pare wood as you'd use a chisel for example then they're definitely not sharp enough. Arguably you should aim for them to be sharp enough that they'll neatly pare pine end grain, although in practice this may not be required for a planer knife. Related: How can I tell if wood turning (lathe) chisels are sharp?
This is giving me a lot of grief as I have the same problem with sharpening hand plane blades,
Although the goal is obviously the same sharpening planes irons can be approached slightly differently. There are some previous Questions on sharpening that might be of some help:
Is there a 'best' way to sharpen an edged tool like a chisel?
How does one aggressively sharpen chisels and plane irons when damaged?
Does it matter what kind of diamond stone I get?
Sharpening grits -- naming and selection
If you'd like further help with your handplane irons start a new thread to get more specific answers on that.
Note on grit size and sharpness: there is no absolute need to sharpen to 1000 grit, although it gives very good results. Many Western woodworkers now sharpen both their chisels and plane irons to well above this but you can actually get a good, workable edge on abrasive as coarse as 250 as long as the sharpening is done well and no wire edge/burr remains.
This is NOT to say the edge is as sharp, it isn't, but it can be sharp enough for the job at hand. For best results though you should go sharper, especially on chisels.