it's not just your local office that's back logged. it's nation wide!
it COULD take no more than an hour for the background check, if you have lived in the same county for your entire life. but if you've lived in multiple counties, states, your local department has to contact each of those counties and run a check on you in those counties. You would think there was a national crime database where all local crimes are reported so that anyone in the country can call one shop and get all the information, but that is apparently not the case.
My sister works at the sheriff's department where we grew up, and she does the background checks for carry permits. She said that she had a guy who was in for his renewal, who lied on where all he's lived. I forget how she found out, but she ran him through the county he lived in but forgot to mention it, and found an assault charge, so she had to deny his renewal when the first time he went through the system, the person doing the checking was just being lazy, and ended up issuing the permit.
When I did my permit, she had her co-worker do all the leg work to avoid any questions that might come up on conflict of interest. Anyway, my background check took her 6 days. Being in the military, I've lived all over the country, and they had to check every county I've lived in since high school. Most of the counties had apparently returned the background check within a day, but Bossier Parish in Louisiana took 5 days to return the background check. If it was not for them, I could have had my permit in 48 hours after filing.
The good news for the OP is that his state law says "SHALL" which means he WILL get a permit, unless he's got criminal history that justifies a denial. so long as his record is clear, his local department HAS to issue a license. Some states laws are written as "MAY" meaning that even if you've never even had a speeding ticket, and are the model of the perfect citizen, the sheriff can still say "na, I don't like you so I'm not issuing you a permit"