No, having a headache and shaking isn't normal. Nor is it, necessarily, related to your blood sugar.
Is it a daily thing that you check your ketones? That's a bit unusual and, if it's part of your standard diabetic routine that does suggest that you might very often be hyperglycaemic.
13.1 is hyperglycaemic and I would certainly be feeling hyper symptoms at that level. If you aren't often in double figures it could be giving you a headache and making you feel unwell. That's because the high blood sugar levels is dehydrating you, so drink some water.
Alternatively, you be waking up so high following a hypo in the night. And that would also make you feel unwell with hypo hangover type symptoms. You can investigate what your blood sugar is doing overnight by doing some overnight basal testing, or if your lucky enough to have a libre or cgm making use of those.
Of coures, you might just have a headache that is a headache and not caused by/related to your blood sugar. Again water might help with that, along with over the counter pain killers.