Hello! With regards to the question of conjugating, both languages, I believe, vary.
Quenya has an easier structure which involves taking the verb na- and, using the example of the first person, adding the suffix -n/-nyë depending on your preference. This makes nan and nanyë respectively. Both mean ‘I’, it just depends on what you think sounds better.
For Sindarin, it is slightly less simple. See, the verb na- is not particularly used apart from its imperative version no! (be!), so conjugating it like you would any other a-stem verb to become non (I am) isn’t what the speakers of the language would have done. Instead, the word ‘ni’ is used. (I will list below the other words for the other persons). This means ‘I’.
The word for a literal doctor of medicine is hadhro, however if this phrase is in the context of Doctor Who, then you may prefer to use the word pengolodh which means a teacher, a doctor of lore which seems more fitting for the Timelord. Next, to join the two words together. You may have heard of prestanneth in Sindarin. If not, it is basically the affection of consonants, causing them to change. For example, to say ‘the forest’ you use the words i meaning ‘the’ and taur meaning forest, wood. However the ‘i’ causes the ‘t’ to mutate to a ‘d’. (This had gone off topic a bit but stick with it!). To say ‘the great forest’ you would then add the word daer, however, because it is an adjective is is mutated to dhaer, therefore becoming i daur dhaer. The copula also relies on prestanneth to show its presence, however more so the lack of it. Eg. to say ‘the forest is big’ you use the same words but without the mutation to daer, therefore it is i daur daer. This links back to your question of how to say I am. You take the word ni and then either word for doctor (depending on your preference) to make either ni hadhro (as oppose to ‘ni chadhro’) or ni pengolodh (as oppose to ‘ni bengolodh’). The words for the other persons are as follows:
*ni I am
*le/ci/de you are (formal/familiar/plural)
*te he/she/it is
*me/gwe we are (exclusive/inclusive)
*ti they are
*se/si this is/these are
*sa/sai that is/those are
*ma what is/who is/who are
As for translating the whole phrase, I will have a stab at it but I recommend that you get the approval of one more learned in the grammar of Sindarin (I will not attempt Quenya for as of now I know not enough) to make sure I have not made ‘a hash of it’.
Estelio nin, ni hadhro/*ni pengolodh.
With regards to estelio (estelia-), I remember this from the films, particularly from the song ‘Evenstar’. However I could not find it on Eldamo nor this site, so I don’t know if it is still an accepted word.