When connected the circuit draws upwards of 10A
I'm not surprised. Consider the transformer primary as an inductor and forget about the secondary or any loading. The primary is still an inductor and you have to get the primary inductance high enough so that the current taken (with no secondary winding/load) is reasonably small.
Primary inductance (in fact it's called magnetization inductance) is proportional to turns squared so doubling the turns makes the inductance 4 times greater and, would reduce the intake of current by 4.
Eighteen turns is never going to be enough. A typical ferrite core (for example) might have an A\$_L\$ value of 10 micro henries per squared turn and this means 18 turns is going to be about 3 milli henries. At 60 Hz this has an impedance of just over an ohm and without a fuse you'd have a fire.
Usually AC transformers have round about 1000 turns on the primary and this of course means a primary inductance of 10 henries and a very much reduced magnetization current. Try looking at this site for some information on transformers.
Also note that transformers do not pass DC because the primary inductance becomes a short i.e. the lower the operating frequency you use the more turns you need to avoid excessive primary (only) current.
If you are still intent on rolling your own, here's the math: -
Also take a look at this medium power transformer: -
Note the number of primary turns. A quick estimation I did tells me there are about 12 layers of about 90 turns each. That's 1080 turns!
Open voltage
This is the voltage across the leads when in that mode. Since the open voltage in diode check mode is only 1.48V, it is unlikely that a LED with a forward voltage much higher than that (usually at least 2.2V) will test properly.
Best Answer
What you have here is a center tapped transformer. The primary winding in this case is the 665 ohm one, and the secondary the 32 ohm one which is split into two windings, so each one giving 16 ohms. I don't know what the rating of this transformer is, but if we assume a 230V/12-0-12V transformer you would get voltage measurements between the windings as shown in the picture. You can find more info about these kind of transformers here: https://www.electronics-tutorials.ws/transformer/multiple-winding-transformers.html
But to answer your question, for this type of transformer all of your readings seem normal for it to be in working condition.
(Image source: 12-0-12 Center Tapped Transformer - Components 101)
(Image source: The Center-tap Transformer - Electronics Tutorials)