When you're on SH 73, the actual exit is posted as SH 82/Martin Luther King -- it's a left lane exit. Take that exit then when you get to the light, hang a right. That's SH 87. You'll be driving right through the middle of the Valero refinery.
Just thought I'd mention that. I can't remember if there are any Sea Rim/SH 87 signs on SH 73.
You can also exit I-10 at Winnie, go south on Hwy 124 until you hit the beach and turn left. You can drive the beach all the way to Sea Rim, if you have 4-wheel drive. Ignore the morons running around naked about 5 miles from High Island. that's also something to remember if you have kids and drive down that way.