How we cite our quotes: (Paragraph.Sentence)
Quote #1
[…] I will only say that I have, with good intentions, contributed towards the organization and administration of the government the best exertions of which a very fallible judgment was capable. Not unconscious in the outset of the inferiority of my qualifications, experience in my own eyes, perhaps still more in the eyes of others, has strengthened the motives to diffidence of myself; and every day the increasing weight of years admonishes me more and more that the shade of retirement is as necessary to me as it will be welcome. (5.1-2)
If you want to retire and other people don't want you to retire, you can always employ some self-deprecation to try and make your case. Washington is basically saying, "Trust me, you guys don't want me to stay." He tries to claim that other people also see his abilities as inferior, which might be a reference to all the division and tension in his administration.
Quote #2
If benefits have resulted to our country from these services, let it always be remembered to your praise […] that under circumstances in which the passions, agitated in every direction, were liable to mislead, amidst appearances sometimes dubious, vicissitudes of fortune often discouraging, in situations in which not unfrequently want of success has countenanced the spirit of criticism, the constancy of your support was the essential prop of the efforts, and a guarantee of the plans by which they were effected. (6.2)
Another way to show your humility is to credit other people for your successes. Like when that group project you did all the work for gets an A+, you say it was a team effort when, really, it wasn't. Instead of taking credit himself, Washington is telling his readers that they, not him, kept the country afloat during all the turmoil of his presidency.
Quote #3
Nor can I forget, as an encouragement to it, your indulgent reception of my sentiments on a former and not dissimilar occasion. (7.4)
Washington is about to launch into the advice section of his farewell address here. But instead of introducing it by listing the reasons why his audience should listen, he instead presents it as them being "indulgent" toward him. As if they are doing him a service, rather than the other way around. It's kind of like buttering them up so that they'll listen to him more, instead of just showering them with sage advice.
Quote #4
In offering to you, my countrymen, these counsels of an old and affectionate friend, I dare not hope they will make the strong and lasting impression I could wish; that they will control the usual current of the passions, or prevent our nation from running the course which has hitherto marked the destiny of nations. But, if I may even flatter myself that they may be productive of some partial benefit, some occasional good […]. (42.1-2)
Talk about hedging your bets. Here, Washington is prepping for people to ignore his advice while also admitting that they shouldn't ignore his advice. What he says in the address is pretty spot-on and relevant to the times, but given what had happened with the political divisions of the era, he seems willing to admit that what he's saying might fall on deaf ears. Yet he doesn't say that they must heed his words or suffer some horrible doom—instead, he just hopes that his advice might be a little helpful…someday, maybe.
Quote #5
I am nevertheless too sensible of my defects not to think it probable that I may have committed many errors. […] I shall also carry with me the hope that my country will never cease to view them with indulgence; and that, after forty five years of my life dedicated to its service with an upright zeal, the faults of incompetent abilities will be consigned to oblivion, as myself must soon be to the mansions of rest. (49.1, 3)
Again, Washington here simultaneously explains why his words should be heeded but also why they shouldn't be. "I have a lifetime of experience, but I'm also incompetent, so don't listen too closely." It's a little hard to tell in writing if he actually feels this way, or if he's humblebragging, but given what we know about his time in politics, it seems more likely that he genuinely doubted himself. He wants to emphasize his efforts more than the outcomes of his policies.