# Do i need independence for paired t-test?

I am testing the mean change in IOP of eyes in a glaucoma study. When looking at just the 'Primary Eye', i am using a paired t-test to test the mean change from baseline as this is one eye per patient. However, when using 'All Eyes' in the study some patients will have both eyes eligible in the analysis (some will just have one eye with glaucoma) and there will be dependence here.

Does the paired t-test assume independence of observations? If so, should i use another test for measuring the change from baseline of patients who have two eyes being analysed?

model <- lmer(IOP ~ 1 + Occasion +(1|Subject)+(1|Subject:Eye),data)