"Saya seharusnya" is awkward.
"Seharusnya" here means "should have", but unlike in English, "seharusnya" is treated as an adverb in Indonesian.
In English context, this sentence-commencing adverb is similar to "Actually, I want..." or "Supposedly, this is because..."
As for your sample sentences, the correct format would be:
Sentence 1:
Seharusnya saya mulai merevisi lebih awal (with the anglicism "revisi")
Seharusnya saya mulai mengkaji ulang lebih awal (with the proper Indonesian word "kaji ulang")
Both "revisi" and "kaji ulang" is interchangeable.
Sentence 2:
Seharusnya saya menerangkan lebih jelas.
In any other sentence, "menjelaskan" would be valid. However, you are correct that using "lebih jelas" after "menjelaskan" would make the sentence very awkward. Hence, you could use "menerangkan" instead.
I hope this helps!