awww :(
try not to beat yourself up. there is no "right" or "wrong" when it comes to stuff like this. you'll get a whole lot of different opinions on how you should wean and what to give them and how to deal with the situation you've described.
personally i went down a baby-led weaning route because it seemed easiest, and my second 2 babies simply wouldn't accept anything from a spoon!
but it's easy for me cos i didn't have twins, and I didn't work, so I could spend as much time as I liked making food, giving meals etc etc. I can imagine you must be utterly knackered most of the time anyway, and tiredness definitely doesn't help when it comes to stressful mealtimes
How are they with finger food? will they eat sandwiches and stuff like that? I think at 13 months none of mine were great with a spoon (marginally better with a fork for stabbing stuff tho), but if they can pick it up with their hands you perhaps stand a better chance?
I tended to just give what we were having, but allow it to cool down quite a lot and then just let them tuck in cutlery or not! it can be messy, but I found it more laid-back letting them eat at their own rate.
you can always then spoon-feed a pudding if you want as well.
I wonder if part of it is them picking up on you being stressed out? I know it's really easy for me to sit here and say "just relax", but... just relax! if they're hungry they will eat the food you offer. and if they refuse then just let them get down from the table maybe?
do you tend to stick to a similar routine at home as they have with the CM?