The sawari sound comes from the interference of the sawari component which may be yama style (non-adjustable) or azuma/en (adjustable). Hereafter, the sawari component will be referred to as either yama or azuma/en for clarity.
The perfect amount of interference is dependent primarily on the tuning - but to be honest, the overall action (height) and tension of the strings are what’s really at work.
With a azuma/en, you can adjust the height of the sawari and thereby the level of interference. So even with dodgy tuning, you can chase a long, mellow sawari.
With a yama, you’re limited to adding or subtracting material such as foil or tape. Because you’re not going to easily be able to adapt your sawari to the tuning, you should pick a key and stick to it.
You sawari should not be too strong, but it should last and last. If your yama or azuma/en is too high, it will be very loud but very short. Remove some material until it mellows and lengthens.
When you have the longest sawari you can manage on your ichi, get your other two strings in tune.
With proper tuning (be careful not to overtighten!), you should be getting some sympathetic ringing. The ichi no ito will always have the loudest sawari, but if you’re getting nothing out of the others, it might be too tight.
Another option is to add a neo-koma. Where the strings tied to the neo meet the dou, you can add a small bar of material to help give them a little more action. This can be plastic, bamboo, bekkou, zouge - whatever. A friend of mine uses tin foil. It shouldn’t be too thick.
One last thing to be aware of:
On nagauta shamisen, the koma should be about 2 finger widths from where the strings flatten over the dou - not 3, as with tsugaru shamisen.