Hi, Betty,
I'm the "DV expert" they were talking about--I've worked professionally in the field for many years.
Just being an alcoholic (or being drunk) doesn't cause anyone to be abusive or to slash tires. Even if the worst behavior is displayed when drinking, the underlying cause is generally a need to possess/control one's partner. Even things like tire-slashing are signs that you could be in significant danger--which also means that your child is, too.
Please contact your local women's shelter and talk with an advocate. If you have sufficient proof that it was he who slashed your tires you can make a police report, but even with much less proof you might be eligible for a protective order that would require him to have no contact with you. The advantage of an order is that the police can arrest for a simple violation of contacting you before it escalates to something more serious.
Even if you don't want to get an order (or aren't eligible for one--in some jurisdictions your physical safety has to be threatened, though in others even harassment is sufficient to get an order), the advocate can help you with safety planning to keep you and your child safe. Please make the call--everything you tell the advocate will be confidential and their services are free.