What do I do if I miss suhoor because of being sick? Do I continue my fast?
Missing suhur is not a reason for not fasting as it is basically your fault. If you nevertheless ate you need to do expiation for voluntary breaking the fast, unless you have a valid reason or for example you found serious hardship in completing the day at that point you might be allowed to break your fast and expiate that day by a day of fast, but only a scholar may give a fatwa on this point after clearing whether what you may describe to him actually fulfils the requirement for such a hardship. As during our fast we should feel the same as poor people who may not have food for days, we shouldn't take the first opportunity to break the fast.
See also Can I fast without suhoor (not during Ramadan)?.
If you are sick and your illness is of a kind that you may need to eat or drink or of kind that not eating or drinking would cause serious harm then you might be allowed not to fast. But this needs both a verdict of a trustworthy physician (Muslim) and maybe a fatwa from a well-versed scholar.
... So whoever among you is ill or on a journey [during them] - then an equal number of days [are to be made up]. And upon those who are able [to fast, but with hardship] - a ransom [as substitute] of feeding a poor person [each day]. And whoever volunteers excess - it is better for him. But to fast is best for you, if you only knew. (2:184)
The month of Ramadhan [is that] in which was revealed the Qur'an, a guidance for the people and clear proofs of guidance and criterion. So whoever sights [the new moon of] the month, let him fast it; and whoever is ill or on a journey - then an equal number of other days. Allah intends for you ease and does not intend for you hardship and [wants] for you to complete the period and to glorify Allah for that [to] which He has guided you; and perhaps you will be grateful. (2:185)