You could try this idea (modified from a Silicon Chip design):

note: You can't really escape the limitation of the PTC fuse on the board though but you do apply power at the same time this way.

And... isn't that FPGA design damn close to a BCD preset 74HC192 counter?

I also recommend that same  IC and it's precisely why I have some in my parts drawers.

I doubt you get better per pin current with either HC or HCT.  HCT was intended to interface the CMOS parts with Legacy TTL (Meaning 5V logic 5V supply).   HCT would not imply higher current capacity per pin.

My Philips datasheet says MAX Package current (Vcc to GND) -is 70mA... so your number has to be wrong.  If each pin were driving a 10mA load, you will exceed *recommended* current for the 74HCT595 device.

No need for PID and all that complexity.

You may have concluded this... but in my opinion,  the PID would serve you well and could eliminate the need for a fan altogether if you do it right.

Quick Note about Hfe

Hfe is one of those things that often confuses someone just starting to ask the question "what is a transistor anyway?"

Another word for Hfe is β (beta)... these odd names are related to the mathematical models used to define how a transistor works... It can also be called "gain".

Gain, being the word least associated with a "math" formula,  is essentially how much does this particular device "transfer" what happens at the Base pin to the other 2 pins.  With "transfer" being one of the base words that make up the word "transistor"... maybe this starts to make more sense?

Well, there is also the mindset that says... "You learn more from failure than success"   So while your experimentation is somewhat beneficial, (you now know more than you did before) you also need to be prepared to replace damaged parts.

I recommend Hakko FX-888.

Buy once, keep a lifetime.

I still have a Weller station that I bought in 1978.

It does sound to be very shift register-like.  Data/Latch/Clock

Certainly... but 1K will neither make the Arduino or the transistor "unhappy"

Using a 1K base resistor for a "typical" NPN BJT with 5V TTL Logic levels is considered to be the de-facto standard as it ideally suited to the most commonly used NPN transistors.  Most general purpose NPN's will have a Hfe rating that puts them somewhere between 150-300 or so. (BDX, being a darlington, has even more gain)   While a notable exception is devices like the 2N3055 Power transistor with a whopping <sarcasm> HFE of 5 at 10 amps, it is typical to "try" using a 1K resistor since it's likely going to have your NPN behaving just fine.

I did install it an use it... and while it "works" nicely, it's really designed for use as a stock room inventory tool in say a corporate or educational or hackerspace environment.  It's "overkill" a few times over for a single hobbyist to manage their parts.

So... I'm still working on mine...

Yes, D3 is clearly redundant, since thats an optical diac in the coupler.

Is there any reason you opted to build your own versus use a Solid State Relay?

You can get a ready to use 10 AMP SSR from MPJA  http://www.mpja.com/, for example, for as little as $7.95.

If you have built the circuit exactly according to the "sample" circuit in the datasheet, then the only conclusion I come to is that your active component (LM317) is broken.  Double check that you have wired it correctly, have use the right components and values... making sure you have specified the correct pin assignment.

I have use this circuit in a home built lab supply since the early 1980's.  The LM317 is very robust if wired correctly.

Well, for 1, why aren't R9 and R10 just 1 20K resistor?  Same goes for the others around the switch that are essentially in series.

To me, that's nonsense.

A 16 MHZ MCU compatible crystal is about $0.50 and add a few more cents for the 2 capacitors needed.  (So maybe $1.00 Max) That's still less than $3.50 you would pay for a crystal oscillator package.