Short answer
To master the /z/ sound, make a long /s/ sound and sing loudly at the same time. It won't sound anything like a J, /dʒ/ at all.
Full answer
The Production of /z/
/z/ is a voiced alveolar fricative. To understand how we make a /z/, we need to think about some different parts of the mouth.
If you look in the mirror you will see a line running down the middle of your tongue (called the mid-saggital line).
If you feel behind your teeth with your tongue, would will feel a little shelf. That's your alveolar ridge. Behind that your mouth suddenly arches upwards to form the roof of your mouth.
To produce a /z/, you have to place the blade of your tongue (that's the bit just behind the very tip) on the alveolar ridge. You create a little furrow or ridge down the mid-saggital line. The rims (sides) of your tongue rest against the inside of your side teeth.
You then force air from your lungs down the narrow channel, this creates friction, or turbulence in the air as the air is forced through this furrow and through the narrow hole created between your tongue and the alveolar ridge. This turbulence causes a hissing noise. While this is happening, your vocal folds (sometimes called your vocal cords) vibrate. This gives the sound pitch which is heard at the same time as the turbulence
It might be helpful at this point to think about how we make the sound /s/.
The production of /s/
To produce a /s/ you have to place the blade of your tongue (that's the bit just behind the very tip) on the alveolar ridge. You create a little furrow or ridge down the mid-saggital line. The rims (sides) of your tongue rest against the inside of your side teeth.
You then force air from your lungs down the narrow channel, this creates friction, or turbulence in the air as the air is forced through this furrow and through the narrow hole created between your tongue and the alveolar ridge. This turbulence causes a hissing noise.
This probably sounds a bit familiar!
How to make a /z/ if you are accidentally making a J sound as in jump, /dʒ/
Now, you will have noticed that to make a /z/ we do exactly the same thing as we do for /s/. There is no difference between the position of your tongue, teeth, or any other part of your mouth at all. The only difference is that when we make a /z/ we have voicing, or vocal fold vibration. This gives the /z/ pitch; we can make a high pitched /z/ or a low pitched /z/. When we make an /s/ we just get a hissing sound. It does not have the same quality of pitch. This is because there is no vibration from the vocal cords.
Because of this, if you can already say /s/ with no problem, you just need to add vocal fold vibration to make a /z/. You need to add pitch. How can you do this? The answer is: you need to sing while you make an /s/. Start making an /s/ and then sing while you are making it. You need to make the /s/ for several seconds. First do it at a high pitch then a low pitch. If you can hear a high or low pitch then you are making a /z/. You can then start practising it at a normal speaking type of pitch.
It is much easier to do this than to try and follow the instructions for making a /z/. If you are making a /dʒ/ sound, the sound in the word jump, then I could give you advice like "move your tongue slightly forward in your mouth towards your front teeth - but don't make a complete closure with your tongue when you start the sound". However, this is very, very difficult to do without anyone to help you. In my experience, singing a note whilst making an /s/ sound is quite easy to do, more effective and more fun.
Hope that's helpful!
References:
You can read about /s/ and /z/ in Gimson's pronunciation of English by Alan Cruttenden, 8th Edition 2014
In connected speech, /ð/ at the start of function words may be assimilated to a preceding consonant in some cases. However, I don't think there are any circumstances where this kind of assimilation always occurs—my impression is that it is gradient. Also, the identity of the preceding consonant probably affects the probability of assimilation.
I have found a source "Applied English Phonology", by Mehmet Yavas, that gives a more specific description of the conditions of this assimilation:
unstressed initial /ð/ in words such as the, this, that becomes assimilated (with or without complete assimilation) to previous alveolar consonants (e.g. what the heck [wɑt̪d̪əhɛk], run the course [ɹ̣ʌnːəkɔɹ̣s], till they see [tɪlːesi], how's the dog? [haʊzːədɔg], takes them [teksːəm]) (p. 67)
I think "takes them" may not be the best example of the phonetic process in question, since them additionally has an alternative form ’em that may occur after any consonant, not only alveolar consonants.
The fact that "in the" could be realized as [ɪnːə] or [ɪnə] rather than [ɪnðə] is mentioned in Geoff Lindsey's blog post "Lucas quiz – the answers".
Another known phonetic phenomenon is deletion (which could be seen as assimilation followed by mandatory shortening) of [θ] or [ð] before the suffix -(e)s. This is lexicalized for many speakers in the noun clothes /kloʊz/, although the non-assimilated pronunciation /kloʊðz/ is not uncommon either. Some speakers (I think a smaller number) also have this type of assimilation/deletion in the word "months", pronouncing it as [mʌnts]. This has been covered in other places on this site (e.g. How to distinguish 'month' and 'months' in pronunciation?)
As far as I know, no native speakers (without speech impediments) use [z] for /ð/, or [s] for /θ/, in contexts other than assimilation to an adjacent /s/ or /z/.
Some native speakers do use realizations other than [ð] and [θ] more generally—I discuss this in more detail in my answer to Do all native English speakers actually pronounce the "th" sound?—but as far as I know it is always something non-sibilant like [d̪], [d̪͡ð], [v]. If you can't manage [θ] in "thorough" or "thief", I would say it's better to fall back on [f] or [t] than to use [s].
As for "at the beginning" and "what the heck", if you pronounce them at a reasonable pace, it will probably not even be noticed if you use a dental stop [d̪] rather than a dental fricative.
Best Answer
First of all, could, good, and cook don't have /u:/, they have /ʊ/, a completely different vowel.
This is /ʊ/: https://en.wikipedia.org/wiki/Near-close_near-back_rounded_vowel
This is /u/: https://en.wikipedia.org/wiki/Close_back_rounded_vowel
Your cuckoo sentence is pronounced:
Secondly, those pronunciation-exercising sentences are called tongue twisters. Tongue twisters are phrases contrived to be difficult to pronounce as you are exposed to it, such as the famous "She sells sea-shells on the sea-shore."
Now, your direct question: "Make a tongue-twister for /n/ and /l/", here's what P.E. Dant said in your comments:
What is "tongue to a sentence" of N & L sound?
The point of using tongue twisters for pronunciation exercises is to be contrived to be hard to pronounce so that you can improve on them, and consequently the sounds the twister contains. I think you can make up your own, as you've already done.