I don't know if your White model allows for the feed dogs to drop, but you could try by setting them to zero. Either way, as long as the stitch plate clicks into place and stays tehre, aligns with the needle hole under there and covers the feed dogs, I would assume it's the right one.
The most important fitting part with the old buttonholers seems to be the presser bar and the screw where the presser fot usually goes. It should of course align with the needle position too, but I haven't come across many variations there. It should be very obvious if it does or doesn't.
Straight stitchers are usually very dependable and if you easily get a thread mess around the bobbin area, I would assume it's either time for cleaning and a bit of maintanance; and it's always a good idea to double (and tripple!) check threading route, needle in the correct way, correct needle type for the machine etc. I have a Singer 99 and a 201, and going from one machine to the other, I have more than once found my self inserting the needle in the wrong way (my 201 threads from right to left, model 99 is the other way around).
If the machine is new to you, double check you have the correct bobbins and needles too. I know I had to read the manual a few times to pick up on all the small things on my 201, and it's a simple straight stitcher too. I think they advice not to mess with the top tension too much, it's not as easy to pick this type appart and get it back together as most tension dials are. Make sure you read up on it in advance at least. My experience is and old machine needs a few rounds of cleaning and oiling before it's back to the way it was meant to be.