Solution for website that exceeds max. database connections

Problem: Website uses MYSQL database. Maximum number of connections to database is often exceeded and site stops functioning for about an hour. What I need: I need a solution, other than using a dedicated server (I'm using shared hosting). I need some type of script or any alternative that you can come up with in order to avoid downtime. Please explain what your solution is before contacting me. If I like your solution to the problem, I will go ahead and start the project. *PLEASE note I'm using SHARED hosting and DO NOT want DEDICATED hosting!