Where exactly is your cadence(s)? You have too many Tonic and Dominant chords with little in the way of secondary chords. It would be so easy to introduce them.
Let's say, you have three cadences of 8, 4 and 4 bars respectively. The general rule, in melody writing, is for 16 bar melodies it is better to either have 8-4-4 division of phrases or 4-4-8 division of phrases (In regards to the amount of bars of each phrase.)
16 bar phrases although not impossible is a bit hard because then you must be able to evolve the melody well and in a meaningful way. It is much easier just to have more cadences. Let us learn to walk before we start debating the finer details of marathon running.
You can start on the tonic, then go to IV in bar two, keep the V chord you have in bar 3, then to something interesting like Vi in bar 4, ii in bar 5, V in bar 6, back to ii in bar seven and then for the first cadence have a imperfect cadence. (ii - V)
So for the first phrase, you would then have the chords I-IV-V-vi-ii-V-ii-V. You have a good amount of secondary chords, you have a proper cadence and your phrase structure is better.
For the second phrase, your cadence can resolve to let's say the vi chord in bar 9, you can have IV in bar ten, ii in bar 11 and then for the second phrase you can have an imperfect cadence again. (ii - V)
For the third phrase, you can now go to the tonic chord in bar 13, IV in bar 14, V in bar fifteen and end on a very good perfect cadence.
For the rhythm, I would not just pick a rhythm at random but choose a rhythm and further elaborate on it. You also need some sort of rhythmical sequence after each phrase.
The use of rest is very unusual. You use them to give a feeling of rest in the middle of a phrase. This breaks the flow of your melody. I feel that those rest are just done for the sake of having rest and not really to improve the melody.
For any melody, there is (generally) many possible chord progressions. Some are better than others, but that depends very much on the style you're aiming for.
As an example, here are two sets of chords that could fit your first four bars
| E - - | E - - | B - - | E - - |
| C#m - - | B - - | E - - | A - - |
Which is better? It depends on the sound you are aiming for.
You do seem to have picked the three major chords (
V) a lot. You could try using some of your minor chords (principally
vi). Try replacing
I chords with
IVmaj7. That is,
Amaj7. Also, try
ii instead of
F#m instead of
I'm also not entirely sure that the
F#m works where you've put it. It sounds a bit strange to my ears. Then again, it might be right for the style you're aiming at.
I would sit down at a keyboard/guitar/other chordal instrument, and just try out different chords. Pick mostly from the E major scale (
F#m, and to a lesser extent
D#dim). You could also try
D out in a few places, to see what it sounds like. Most will probably be average, but you'll find a few good ones.
There are lots of other questions around here about choosing chords. Perhaps have a read through a few, and see if you get inspired.
This sounds like the style is basically Indian, and I think the reason you are having trouble with the chords is because you are trying to "westernize" it but it doesn't want to be westernized!
Leave it the way it "wants to be." Play it over a drone, with a few notes in a middle part if you like, and some percussion.
(If somebody wants to complain that "this doesn't answer the question", that's OK - I'm not bothered about the rep, but I'm not going to set up a dropbox account just to put an link to the image in a comment!)