Hi Katrina
Personally i agree with everybody else here and say the dose given by your vet is too high. When my Duke started on Prozinc in January the vet gave us a starting dose of 1.5U and that was too high and he nearly had a hypo on the first day so we had to reduce that and build up slowly to his required dose. Looking at Tuna's SS his numbers are very similar to what duke's were when he started on his insulin.
It's easier to start on a low dose and build up rather than risk a hypo and bouncing all over the place.
As for syringes i use Prozinc U40 syringes that have 1/2 unit markings, i have found this one in the USA but there will be many other sites that sell them as well.
https://www.valleyvet.com/ct_detail.html?pgguid=fa6db1a3-6eac-45d4-a304-d395c5a01dcb
Just one more thing with the syringes when you are using them, check the syringes before you use them as they can vary slightly as can the U100 syringes.
See this post where i had problems with syringes.
https://www.felinediabetes.com/FDMB/threads/help-regarding-dukes-numbers-please.243125/#post-2744357
I just sort them all out before i use them so i'm not messing about at shot time, three separate bags with the plunger at the top, middle and bottom of the zero line, i then use all the syringes with the plunger at the top, then the middle and then the bottom. This way i can give more or less the exact dose each time, and i adjust the dose very slightly when i use the ones where the plunger goes to the middle and bottom of the zero line, but still the same dose if you see what i mean.