It'd be very wise to go ahead and be honest about it with your shrink (I just like the word; I don't get them overinflated in my mind LOL). What's the reason that you haven't yet? It's not necessarily going to land you in the hospital; they'll ask you questions to find out to what extent the suicidal thoughts are going on. So long as it's not an emergency situation, you'll have the opportunity to have more/better/more appropriate treatment, sensitively, and you needn't worry.
If you're really in crisis, the hospital isn't necessarily a bad thing, itself. Most people don't think it's the most fun thing in the world, but on the other hand it's kind of a temporary refuge from the stressors one's been dealing with, so the experience sort of evens out. They usually only keep a person for a week or so; while making sure you get stabilized & are getting a better outside support system going. It's extremely rare for a person to be forced into a hospital, because the system to do so is very rigorous. Virtually all hospitalizations are voluntary.
One time I was really suicidal & went into a mental hospital; was there for, I think, 6 days. One thing I can say about it is that it really galvanized me into making sure I had a support system set up for when I left (plenty of therapy, added couples' therapy - maybe a certain amount of family therapy would be possible/helpful for you, don't know). I'd had a very mediocre psychiatrist before going in; when I came out I was about to tell him I was going to find someone else. Before I could, he informed me that he was moving out of state. Lucky me!
What I did to find someone good was to ask everyone (my other docs & therapists, including my family physician, who they'd heard of that people liked). Several people mentioned the same name, and I went to her. Major improvement. This is a little off the primary topic, but illustrates the need to advocate for yourself at times & how to go about making a good change. This is a bit of a ramble, & I apologize, but if someone wants to pm me about it I'll be glad to go into more detail with you about any aspect of finding a good doc, hospital, etc.