The second part of the dream sounds more like a projection with someone else to me (I could be wrong, there is some symbolism to it).

A baby in a dream has many possible connotations- it can refer to your nurturing instinct, and also to your distrust of other mothers (the mother let you deal with the baby instead of coming to your aid and taking responsibility)- but I get the feeling that the baby is also symbolic of your budding spirituality- letting go of fears and doubts (the baby was going to your left side, and this would be the intuitive side) may be too much for you to take, and you want to ease this burden.
So there's a few ways to look at this.