A couple of questions.
- Blazer or pickup?
- ControlTrac (2wd, 4wd, 4wd-Lo) or AutoTrac (2wd, Auto 4wd, 4wd, 4wd-Lo)?
- If you have push button 4wd, does the light continue to blink after 4wd is selected?
If 4wd is not engaged on a regular basis, the Transfer Case Control Module (computer) will lock up. To reset it, disconnect the wiring harness at the TCCM (behind right kick panel) or disconnect the battery for a few minutes.
The vacuum switch that engages the front axles failed on my 2000 S10 Blazer. Fortunately, it is located on the firewall. I bypassed the switch by disconnecting the two vacuum hoses from the switch and splicing them together. This enabled the actuator (located under the battery tray) to engage the front axle. Check the vacuum hoses for rot, I've had to replace a few after 10 years.
When 4ed is engaged do you hear any noises from the transfer case? The transfer case control motor is mounted on the transfer case. I can't offer any suggestions on this one.
Here's a link that helped when I was troubleshooting the Blazer's 4wd.