Read up on Depth of Field, and 'selective focus'. your problem is a combination of issues, but your disappointment with the 35/f1.8 I suspect is largely not understanding what it's strengths are.
But first, 'Infinity Focus'. This is the distance beyond which a lens will make pretty much everything, no matter how far from the camera 'sharp'. Most camera lenses, though offer a range of 'variable focus' shorter than that, starting from a 'closest focus' point.
Depth of Focus is the zone of acceptable sharpness in-front and behind the point you actually focus on, and is a % of the focus distance. So, in theory, if you have a large Depth of Focus, and a relatively 'near' infinity focus distance; you can almost have a 'focus free' lens, the Depth of Field so deep that it starts close enough to the camera, and extending so far away, everything you can see is 'acceptably' sharp. But, where focus distance is within the 'variable focus range' and the Depth of Focus doesn't extend to infinity focus, you get a 'Focus Zone' with near and far subjects loosing sharpness.
Now, with longer focal length lenses, their nearest focus distance, and their infinity focus distances tend to go further away from the camera, and so the range of variable focus between them, becomes larger, while the 'depth of focus' for any given aperture tends to become smaller.
Meanwhile, a lens is a lens no matter what camera its mounted on, and its focal length, near and infinity focus and the DoF it offers are proportional to that focal length. BUT, if we put a smaller 'frame' or 'sensor' behind the lens, the picture you get in that frame, is 'cropped' and gives a smaller angle of view than if the frame were larger.
So, if you want really sharp photo's, 'everything' no matter how near or far from the camera rendered in 'sharp' focus, you can exploit this idea of a 'focus free' lens, by using an incredibly small frame or sensor, that gives a 'normal' angle of view from an incredibly short focal length, that inherently has a very very close focus distance and tends to infinity very near the camera; and in fact, this is exactly what 'camera-phones' often rely on, rather than a mechanical focus mechanism.
BUT, if you don't want photo's with total front to back sharpness, you actually want a shallow depth of focus to be able to throw distracting back-grounds out of focus and emphasize the subject, then, smaller sensors and shorter lenses aren't so helpful, and but while they have larger sensors than camera-phones, 'consumer' DSLR's do have smaller sensors and shorter lenses, and while you can achieve 'selective-focus' effects with them, due to the added 'zoom' of the crop-factor, you have to stand back further, increasing the camera to subject distance, which inherently will increase the Depth of focus too, or you have to get up closer and use a wider angle lens.,., that inherently offers greater depth of focus as well... SO the only other way to compensate and reduce the depth of focus and get a more exaggerated selective focus effect is to open up the aperture... which on most 'kit' lenses on consumer DSLR's are frequently quite restrictive... typically around f4.5/5.6 or so. Making 'faster' (larger aperture, lower f-number) lenses, though, is expensive; The Nikon 35/f1.8, then was designed as a low-cost crop-camera lens, that sacrificed 'zoom' for a fast maximum aperture, which has a few other advantages, but being able to more readily utilize it to explore shallow DoF effects, where the smaller sensor and shorter focal lengths are working against you, is chief amongst them.
Meanwhile, at close subject distances, DoF can be very shallow, even at moderate apertures, f5.6 or f8. With a 'normal' angle lens, Camera to subject distance to 'frame' a head and shoulders may be as little as 'arms length'.. I have long arms, so for me that's about 1m, and at f8, on a 35mm lens, the DoF would extend from about 75cm to 1.5m, At f4, from 90cm to 120cm, a 'range' of just 30cm, 'just' enough to get nose and ears 'sharp'... BUT open up to f1.8? that DoF falls to almost nothing, a few mm either side of the focus point.. NOW you focus on the eyes, and the nose and ears are blurry.
So, even if you nail your focus, working that close, that wide-open, you are likely to get such a shallow DoF that very little is 'sharp'... it's a feature of the lens, not a 'fault'.
And the fact that you say that your shots were 'disappointingly un-sharp' sort of implies you are missing the point of it, that fast aperture is significantly there so that you can make photos significantly un-sharp... but where you don't want them to be! And discovered the 'tricky bit' getting the sharp bit where you want it! Focus IS critical, but, understanding how DoF works, and how much you need and how to get it is more so. Getting to grips with the focus modes is only part way to solving the problem. Knowing how shallow the DoF will go as you get closer and as you open up, or how deep it will get as you move away or close down, is the key.