agree with Marionk
If you're a nice person, which I think you might be because you are insightful about your feelings and worried about them, FAKE IT till you make it!
My mother reacted to the birth of my second by OTT spoiling my first, saying it was so that she didn't "feel pushed aside by the baby". It was awful not just for me as a new mum (whose baby was being ignored) but also awful for DD1, who had/has a great bond with DD2 but found her grandmothers very obvious favouritism confusing, and she began to act out towards the baby after Nanna had been to visit (she didn't usually act out to her, I think she was copying Nanna as someone that she looks up to, she could see Nanna making a point of turning all attention away from DD2 and onto DD1)
It never improved, we ended up seeing less of her because (amongst other reasons) she may think she was doing DD1 a favour by treating her "specially" and not making a fuss of the baby, but actually it confused DD1 (who was otherwise besotted with new little sister) and I didn't want it to affect DD1s fabulous bond with DD2..
Parents-in-Law. What do/did you call them?