I notice that a lot happened four months ago. You were denied credit twice. Your income went up from $20k to $60k. I'm wondering if you were denied credit based on your $20k income. Since you couldn't provide proof of your income I wonder if they used $0 for your income.
Debt to income ratio is one significant factor included in the credit score calculation. You may not have a lot of debt, but if you don't have any income even a few hundred dollars on a credit card would throw your debt to income ratio into a panic.
I'm assuming that your change from $20k to $60k income involved a change of jobs. Perhaps now you can provide proof of income. You would certainly need to do that before being approved for a mortgage.
Well that's my two cents about what may (or may not) have gone wrong last time. As for what to do next I would agree that the most helpful thing you could do is check your credit score and fix any errors that might negatively impact your credit score. (There might also be non-errors that need addressed such as open credit accounts that you thought you had closed.)
When building credit history, time is on your side. If you just go on living your life and paying your bills promptly, your credit will slowly climb to an acceptable level. Unfortunately in the time frame you mentioned (~1 year) there isn't really enough time to build it significantly.
You bring up a valid point about credit applications reducing your credit score. Of course, that effect is somewhat minimal and temporary (2 yrs according to the thread linked to above). But again 1 year is not enough to recover. If you're considering applying for additional credit as a means to improve your credit score it may be too late to reap the benefits before your mortgage application.
Of course if you could pay off any debts, that would help your debt to income ratio. But it would also reduce any house down payment you could save up and thereby increase the amount of your mortgage.
Better just save those pennies (or preferably Washingtons and Benjamins) to put toward a down payment.