Translation:My dog keeps getting sick.
生病 is “to get sick/to develop a sickness”. This is why we translate present tense “to be sick” when the Chinese 生病了 uses perfect aspect (“got sick/developed a sickness” --> “is sick now”).
The difference between 总是 and 一直 is difficult to define… 一直 feels more like “all the time, continuously, perpetually”.
It is not always ill, it keeps getting sick - for example it gets sick every time you want to go to the theater - that is 3 times a year. "I never get to the theater because my dog keeps getting sick" - but this dog is pretty healthy all the other 362 days of the year :)))))