Your total credit card limit is $3,700 and your maximum installment loan outstanding was $3,800.  If you open another installment loan for the amount of $5,000 that would seem to boost the total credit limit by about %50, while installment loans count much less against utilization.  So, this should boost your score.

Having two lines and paying on them regularly without lates should improve your score more than using the money and paying off the first loan (which will cause it to close and thus start the aging process until it falls off of your report).... so if maximizing your FICO is the goal, I think opening the second loan should be effective.  (I've got a similar questions in a thread I created, but so far havent' really gotten a clear answer.) 

The FICO simulator I use says that boosting my total credit liimt bt %50 boosts my score dramatically (everything else the same) And it is counting everything as "credit limit" (including things that are paid off.) 

BTW, it appears most companies report to the credit unions the amount on the statement.  Thus if you pay the day after the statement your utilization will be higher than it would be if you just paid the day before the statement.  IF you have online access to your accounts you can pay the balance the day before the stament day and this will bost your fico scores by lowering your credit utilization.  (you could even set up automatic payments along these lines.)