Wash your hands thoroughly after every trip to the bathroom and take some sanitising wipes and wipe down the light switches, the TV remote control, the phone and the door handles in your room if you think this would make you happier....some people carry this to extremes and refuse to use the pens the waiters hand you to sign the drinks chits ( and only use their own) and some use sanitisers to pick up the utensils in the buffet area....you can go overboard with worry if you think about it .
I have just returned from a caribbean cruise with sea princess and I have to say I don't think you have anything to worry about. I am a very particular person as regards cleanliness and have very high standards and I was pleasantly surprised. Our cabin steward was very thorough with the cleaning and the accommodation was lovely. All public areas were cleaned and maintained on a daily basis and I found no evidence whatsoever of poor hygiene. All I will say is that any problems on board ship have been brought on by passengers whose standards are not so high! Entertainment was good the first week but did lack a little in the second I must admit. It was our first cruise and like you we were concerned as to what we could expect, but there was loads to do both in the evening and the daytime, it just depends how much you want to join in.
Entertainment is difficult because everyone has different ideas of what they like. However Sea Princess is a lovely ship with great atmosphere. I am sure you will like Alan and Alana (brother and sister Piano/singing act)who run very popular quizes as well and are delightful people. She also runs dance classes as she is an ex dancer. The public areas and toilets are cleaned regularly throughout the day and hand cleanser is available at all times. However many, especially men, do not use them. The ship tries very hard and it is up to all to encourage others to practice good hygene. You will have a lovely time.