It looks like your translation is spot on.
This answer is based on the site Ardalambion .
In translating "I am free. I am myself.", the tricky part was "I am". It appears to be one word, "Nanyë". I found this explanation for the word:
With pronominal endings this verb may appear as násë "he/she is", nás "it is";
otherwise the regular pronominal endings are added to na- (nanyë "I
am", nalyë "you are" etc.; see full list of pronominal suffixes
below). The past tense "was, were" would be similar, only with the
vowels e, é where the present tense has a, á: hence nenyë "I was",
nelyë "you were" etc.
Reference Bold mine
Which is about as clear as mud. However, what it mean is this: na- is the present tense "to be" verb, and by adding the suffix -nyë (which is the pronomial suffix for "I") you make the new word nanyë which means "I am". (In Quenya, "I" is rarely an independent pronoun. "I am" would look like "amI", as one word.) (And yeah, it's weird, but I guess J.R.R. really liked Finnish.)
So, the first word is correct.
The second word and the fourth word are both in this dictionary.
The second word "free" is in bold:
FREE (adj.) léra, aranya (not to be confused with aranya *"my king";
the shorter form ranya also cited must not be confused with the verb
The third word "myself" is in bold
MYSELF (reflexive pronoun) immo (a general sg. reflexive pronoun,
covering English "myself, him/herself, yourself").
The third word is the first word again.
I think "I am free. I am myself." is "Nanyë léra, Nanyë immo" in Quenya.
The Tengwar translation, is great, but there are a few things I think you should check. I am using this chart and these rules.
First word, "Nanyë": Everything seems correct.
Second word, "léra": Wow, you got that the long e needed a carrier! That's amazing!
However, the last letter is "a". The vowel symble over the "r" character (looks like a long y) is a slash, which is "e". It should be three dots, which is "a".
Third word, "Nanyë": Should be the same as the first word, but the vowel symbols are missing on the last character.
Fourth word, "immo": Great job getting the line under the "m" to make it a double consonant. But the "o" after the double "m" is not indicated. There should be a crescent mark above the double character to show the "o".
OK...now that I'm at work, here's how I think the Tengwar should look:
Yep...your new edit is spot on. Short carrier for the initial "i" and the "n" carrying the "y" and the "e".
I hope that is helpful!