Focusing only on single vs double butted spokes: I am not sure that the distinction is very important in your case.
In the bicycle industry, butting means that the wall thickness of a tube or the diameter of a spoke varies across its length. Usually, the ends will be thicker, since that's where you need more material - this applies to frame tubes and spokes. Single butted means that the tube is thicker only at one end, and double butted means that it is thicker at both ends. Triple butted tubes exist, and I believe that for frame tubes this means the thickness is further reduced in the center.
All the major spoke manufacturers offer straight gauge spokes, which have constant thickness, e.g. 2.0mm thick is common. Double butted is also common, e.g. 2.0mm thick at the threaded end and at the j-bend, 1.8mm in the center. This previous question discussed the pros and cons of double butted vs straight gauge spokes. I am not sure if you meant double butted vs straight gauge or not.
Single butted spokes do exist. One example is the Sapim Strong, which is 2.3mm thick at the j-bend, and 2.0mm thick everywhere else including at the threaded end. (NB link goes to a commercial site and may not be durable; the principle should remain and I believe that DT Swiss, another major manufacturer, should offer equivalent spokes, as well as other spoke manufacturers.) The asserted benefit is that this is stronger and more suited for heavy duty applications like tandems, e-bikes, maybe really heavy duty MTB applications. I'm not a wheelbuilder, but I think many breakages that are not a result of poor building or spoke quality control may occur at the j-bend, so if true it could make more sense to put more material there.
Additionally, I have a feeling that most single butted spokes may be 2.3mm at the j-bend. From my perspective as an observer, I did not see spokes with that diameter in use with rim brake road bikes during the 2000s to 2010s. I have a feeling that extra strength is unnecessary. Disc brakes impose more torque at the hub, so it could be sensible to have spokes that are 2.3mm at the j-bend in that case - but I don't think that diameter is universal on disc brake drop bar bikes either. I am including this commentary about rim brake road bikes because that is what the OP has.
More generally, as discussed in the previous answer, you could just use straight gauge spokes, which are cheaper. They may be less comfortable, and the conventional wisdom is that they are less durable than double butted. However, a big part of a wheel's durability is how many spokes compared to the rider + bicycle + load weight, how strong the rim is, how straight the rim was when it was delivered, how well the wheel was built. Chances are that 36 spokes are sufficiently strong if the wheel was built well. If you went with double butted spokes, there may be a theoretical durability advantage, but it may not matter in practical terms. For a rim brake bike, the sense I get is that as long as the spoke is 2.0mm at the ends, it's strong enough (quality factors discussed above aside).
As to how to tell how thick your spokes are, you could obviously use a pair of digital calipers to measure the spokes at the center and both ends. If you have the same thickness throughout, you have straight gauge. Same thickness at both ends but thinner in the center = double butted. Thickest at the j-bend but same diameter everywhere else = single butted. I am not sure if non-digital calipers will suffice suffice for this, because they may only have a precision of 0.1mm. In any case, I don't think it is important to measure the diameter of your current spokes; you need to know their length, and you can select a double butted or straight gauge spoke at your leisure - and yes, you could choose to use a single butted spoke for disc brakes if you want, I just don't see a strong rationale for this.
