I work full-time and am doing a Master’s too
I work full-time and I am doing a Master’s too
Are both the sentences correct?. What is the difference
between “am” and “I am” in the context of sentences
above?.
Neither is correct as they don’t have full stops! 
Other than that, they are both the same. The only difference is that in the first sentence the ‘I’ is assumed to belong to both verbs. This obviously makes the second option the more precise one.
in english sentences are organized in a “SVO” structure meaning subject, verb, then object. Usually the subject is in a noun phrase, the verb in a verb phrase and the object also in a noun phrase. your sentence is broken up with a conjunction so its like you have two sentences and each of these sentences need to use the SVO structure. So it should be “SVO conjunction SVO”.
Your first sentence is “I work full-time”. The subject is “I”, the verb is “work” and the object is not mentioned but it is assumed. You use the word “full-time” which is an adjective describing work. This isn’t describing the former verb work it is describing the absent noun work. But just like “play” is assumed in “I play full-time” so is “work” assumed in “I work full-time”. You don’t say “I work full-time work” or “I work work” because the verb you use makes it clear what you are doing. So the object is “work”. In the second sentence “I am doing a Master’s too” the “I” is the subject, “doing” is the verb, and “a master’s degree” is the object. Now “degree” is left out but it is also assumed here.
Since both your subjects are the same you can omit the subject when it is repeated because you already introduced it and you don’t need to do it again. This makes it look like the structure of the sentence turns into “SVO conjunction VO” and by looking at the words only that is true but the subject is assumed in the second sentence and so it still follows the SVO structure.
I can do the same thing with the verb and the object as well. For example “I laugh at good jokes and they at bad ones”. “Laugh” is the common verb here but I omit it the second sentence but it is clear that laugh is the verb. For the object I could says “I love pizza but he hates it” In the second sentence I simply use “it” but it is clear that “it” means pizza. I can also combine them saying something like “I help people but he doesn’t” In a long form this would be “I help people but he does not help people” However I don’t need to use the verb “help” and the object “people” in the second sentence because they don’t change so I can omit them.