If all of the following conditions are satisfied:
- Your utilization is consistently high due to a very low limit relative to your monthly expenditures
- You pay your balance every month
- Your income and score is high enough to warrant a credit limit increase
- There are no errors in your credit files
Then applying for a credit limit increase will most likely result in an increase in your score, because the high utilization will have a greater impact than a single hard inquiry that the limit increase request will generate.
But if your current score and reported income is not high enough to warrant an increase, the hard inquiry will lower your score slightly. The issue here is whether the creditor thinks you could handle a higher limit, and this is significantly related to what annual income you have reported to them, as well as any other debts that they might see in your credit files. A good history of timely payment on their own account is also a factor, but not the only one.
With respect to discrepancies in the scores, note that CreditKarma reports TransUnion and Equifax credit files and scores, which are based on the VantageScore 3.0 model; Experian uses FICO. These different scoring models should be in the same ballpark but it is not unusual for there to be variations. Also, different agencies may report different information--for example, hard inquiries are generated on different files because a given creditor may request your file from one agency and not the others.
On a personal note, I myself was once in a similar situation. I distrusted credit offers (after having been burned by predatory lenders years ago), and the only card I had was issued by my bank, with a very low limit. I used it constantly, but always paid it off. Eventually I realized that I was not doing myself any favors:
- My utilization was all over the map because it depended on whether I had just paid off the balance before the creditor reported it to the agencies
- I was not taking advantage of card offers with rewards, which I rightfully earned by not carrying a balance
- Limiting myself to a single card was risky since it is a single point of failure; if it were lost or stolen, the inconvenience would be substantial.
So, I applied for a few lines of credit, and increased my total limit by a factor of about 30, dropping my utilization from nearly 100% each month to less than 3%.