You have a few problems. First, you say the LEDs are 20mA standard? Yet you list 15Ω resistors. That's ((5 - 3.6v) / 15Ω) = 93mA EACH. They won't last long.
Second, transistors are not just on/off. That's only if you use them as a switch, in saturation mode. They can and are designed to have a wide range of use.
Third, when SW1 is opened, you leave the base pin and resistor floating. It's neither 0v, or 5v, ie "Off" or "On". It can float between the two, and cause the transistor to switch on. Basically like a little antenna.
Fourth, you have the floating pin and resistor on a breadboard, which has high capacitance. It acts like a capacitor, storing then releasing energy when it can, which is why it varies when you bring your finger near the resistor. Again like a little antenna.
Try adding a 10k resistor after the 100Ω resistor, to ground. This creates a weak pull-down, which will turn off the transistor unless the SW1 is closed, which will override it.
As for the specific IC you linked to, they essentially do the same. Instead of a NPN transistor, the "typical use/reference design" they provide uses a N-Channel Mosfet, but includes a 180k pull down (lower left). A NPN like you have will work just as well, BUT they don't list a maximum current that can be sourced from the load pin. I'd assume, based on the Backlight pin (tested at 3mA at 5v with a max 0.5v voltage sag, see datasheet page 17), it's only a few mA, probably 5mA at best. You might need a high gain transistor, or a darlington pair.
![enter image description here](https://i.stack.imgur.com/QB6F3.png)
None of those transistors (nor the TL431 regulator) are particularly suitable.
Look for an NPN silicon transistor with an ft of 600-800MHz. The best fit is the high-voltage 2N5401 with an ft of 300MHz, but at 100MHz+ FM it won't be that great.
Best Answer
BC331 NPN, 60v, 30 mA, hFE= 125, Ft = 100 MHz, TI, low noise, TO92
Very old and tired.
Some data here
Related selection guide here
BC541: Does not seem to be a known device.
A BC547 will be a good substitute for most older silicon NPN transistors.
Note that BC541 = 60V rated and BC547 = 50V. This is extremely unlikely to cause a problem.
For most small signal transistor purposes I use and recommend the following:
BC337-40 NPN through hole Datasheet here
BC327-40 PNP through hole Datasheet here
BC807-40 PNP SMD Datasheet here
BC815-40 NPN SMD Datasheet here
All 45V, typically 0.8A (varies slightly with manufacturer), TO92 or SOT23 Ft = 100 Mhz Hfe (current gain = 250-600, mean = 400.
All these are often available as cheaply as any other transistor available (look around) and have abpout as good a combination of current, current gain and general usefulness as anyhing else going. When you find better cheaper please advise.