I would try to resolve the registration issue first, if you aren’t registered, you wont be able to reliably send and receive data.
If you are registered the light should blink once every ~3s. (every 800ms means not registered, every 300ms = connected to GPRS and tx/rx data).
Did you set the apn correctly? For reference the series of AT commands I used to initialize the SIM800H/SIM800L are:
"ATE 0\r\n" //disable echo
"AT+IPR=115200\r\n" //hardcode serial baud rate
"AT+SAPBR=3,1,\"CONTYPE\",\"GPRS\"\r\n" //use GPRS for IP based bearer
"AT+SAPBR=3,1,\"APN\",\"hologram\"\r\n" //set APN for IP based bearer
"AT+CSTT=\"hologram\",\"\",\"\"\r\n" //may also want to run this to set APN
"AT+CFUN=1\r\n" //set modem to full functionality
//then periodically run:
"AT+CREG?\r\n" //check if registered
"AT+CSQ\r\n" //check signal quality first number in response should be > 5 for "ok" signal
This may not fix the SMS but is all I can offer as I don’t send SMS from my device but you definitely need to fix the registration problem before tackling SMS.
Also make sure you have the antenna properly connected and use a decent antenna, the tiny ones that come with most SIM800L kits are complete crap. I’d spend like 10-15 bucks on a GSM dipole antenna from amazon or ebay, don’t get one with an integral long RF cable, these RF cables are also crap and the cable loss will attenuate too much signal. Or, if you re ok with shipping being more than the part itself, I use a Laird EPR8221A1-15UFL and have good results.