Vikram, I used to have EF 100-400L IS, which is a great lens but I sold it and got Sigma 150-500 OS for better reach and I found the push pull to be a bit awkward ... my copy of the Sigma 150-500 OS is just as good as my earlier 100-400L IS lens if not better.
The Sigma 150-500 OS sells for about $1000 (http://www.bhphotovideo.com/c/product/549255-REG/Sigma_737101_150_500mm_f_5_6_3_DG_OS.html) which is well within your budget, giving you enough money left over to buy:
1. a good quality filter
2. a Lens Coat Lens cover (http://www.bhphotovideo.com/c/product/652524-REG/LensCoat_LCS150500M4_Sigma_Lens_Cover_Realtree.html)
3. an external flash
4. a Better Beamer flash extender (http://www.bhphotovideo.com/c/search?Ntt=better+beamer&N=0&InitialSearch=yes&sts=ma&Top+Nav-Search=) this will come in handy to illuminate the birds so your images get a nice separation and pop
Here are a couple images made last week with the Sigma 150-500 OS (using an external flash + better beamer)
Nice pics. I like the catchlight in the eyes.
Thanks for your kind words. Cheers