Learn Russian: Я сам очень ценю преданность и искренних друзей. - I myself highly value loyalty and sincere friends.

The verb ценю (I value) requires the accusative case. Since friends are living beings (animate), their accusative plural form matches their genitive plural. The word друг has an irregular plural друзья. Nouns in this special -ья plural group usually take the ending -ей in the genitive/accusative plural, giving us друзей.

Why is the adjective искренних used instead of искренние?

Adjectives must match the noun they describe in case, number, and animacy. Because друзей is in the accusative animate plural (which borrows its form from the genitive case), the adjective искренние (sincere) must also take the genitive/accusative plural ending, which is -их.

Why doesn't the word преданность (loyalty) change its ending, since it is also a direct object?

преданность is a feminine noun ending in a soft sign (-ь), meaning it belongs to the 3rd declension. In Russian, the accusative form of 3rd declension nouns is identical to their nominative form, so the word does not change when used as a direct object.

If I am a female speaker, would I still say Я сам?