I can relate to your feelings. Although I do want to get better and have been working very hard at it for years, I have been obsessed with constant thoughts of suicide since I was a kid.
I agree that you should share your thoughts with your T. I try not to but mine requires me to turn in a DBT diary card at each session which ranks my SH and SUI thoughts or actions each day among other things.
He says he needs to know and we need to work on that first becauae it will prevent me from making progress. Although I hate the diary card, I do kind of like it for sharing those things because I can just put down a number each day and not have to say anything unless he brings it up.