Yeah, I'd say the vast majority do not have showers. Most places I've worked in the LA area don't.
That said, here's a few other ideas:
1. Check the local area for small health clubs or other venues of exercise. There's a rock-climbing place near where I used to work and I used to pay them $5 a month to use their showers. They were about half-a-mile from the office, so after showering and changing, I'd hop back on my bike and ride in to work.
2. Where I work now, that wasn't an option. So in the office, I have two sponges, face-cloth and hand-towel, soap, coat-hanger, and a large cup. I ride in with clean clothes in my backpack. When I get to the office, I grab my stuff and go into the bathroom. I soap up one sponge and clean myself up--wash under the arms, etc. I take the other wet sponge and wash off the soap. Pour water in my hair, towel everything off, maybe throw on some body-spray. I throw my sweaty bike shirt in the sink and wash it, wring it out, and stick it on the coat hanger. Same with the bike shorts. I change clothes and come in with the wet clothes, which I hang up behind a server to dry them out.
3. Speak with an outrageous French accent and everyone will think the smell is normal. (Sorry French people, couldn't resist...)