There are local companies that hire recent CDL school graduates. You just have to beat the bushes. If you want to do OTR for a year....go for it. If you want to explore other options...keep your eyes open (i.e. Food service, LTL, etc.).
On the medical card...don't be shocked if the company you eventually go with makes you go to a "company" doctor for another DOT physical. When I got hired on...I had to go get another DOT physical...two months after my first one, lol.