have you got a baby swing?? i higly recommend one, or a vibrating chair, we would be lost without ours... also you can buy any type of sling thru internet and get it delivered in days...
not that this helps but at 3 i think its fine for them to have intermittent supervision in the bath.... just don't make it overly deep... i am encouraging our 3 year old to have a bit more alone time in the bath as we also have a newborn (well 11 weeks now) so can sometimes be hard in the evenings...

it definitely gets easier as your newborn gets more settled though, hang in there....