I do not believe that the verses cited mean that it is easy to live a Godly life, and to abstain from sin simply because we're saved. No less than Paul himself, who wrote a good portion of the New Testament struggled against his sin nature. He wrote about it often, but I think that the most raw and honest writing, which dealt directly with his struggle is found in Romans 7. (Quoted from the KJV)
15 For that which I do I know not: for what I would, that do I not;
but what I hate, that do I.
16 If then I do that which I would not, I consent unto the law that
it is good.
17 Now then it is no more I that do it, but sin that dwells in me.
18 For I know that in me (that is, in my flesh,) dwells no good
thing: for to will is present with me; but how to perform that which
is good I find not.
19 For the good that I would I do not: but the evil which I would
not, that I do.
20 Now if I do that which I would not, it is no more I that do it,
but sin that dwells in me.
21 I find then a law, that, when I would do good, evil is present
22 For I delight in the law of God after the inward man:
23 But I see another law in my members, warring against the law of my
mind, and bringing me into captivity to the law of sin which is in my
24 O wretched man that I am! who shall deliver me from the body of
25 I thank God through Jesus Christ our Lord. So then with the mind I
myself serve the law of God; but with the flesh the law of sin.
In summary, Paul is saying "I want to do good, but I don't do it, and I HATE it!" (verse 15). Then he goes on to tell about how in his flesh, his earthly nature, there is no good thing, but he also delights in the Law of the Lord, and there is a war going on inside him, constantly, with his flesh warring against his spirit.
If Paul, who was willing to suffer imprisonment, torture, and death for the cause of Christ had problems living Godly, then who are we to expect to just sail through it?
In my personal life, I can answer this way (admitting up front that this is subjective, and personal). I had a lot of things in my life that were sinful. I won't list them here, but there was a list. I didn't give them up immediately when I was saved, and when I tried to give them up under my own power, using my own willpower and determination, I was unable to escape the bondage of my sins.
Eventually, I learned how to simply give up and ask God to make the changes in me. Being a proud, stubborn, willfull person, this was a very hard lesson to learn. But the truth is, once I asked God to take away my desire to do certain things, He did it. Not immediately, but He did.
Taking just one example, I was into pornography from the time I was 15 or so. Giving that up, especially with the Internet at my fingertips was difficult. Like Paul, I knew what it was like to want to live one way, but keep getting sucked into sin. It was like quicksand with tentacles that reaches out and grabs you. Finally, I learned to give up and stop trying to do it under my own power, and just gave it over to Him, asking Him to remove the desire, and as I said, He did.
When I read about the load being light, and finding rest, I think about that struggle and others like it. Once I turned it over to Him, it was easy, and the internal warring eventually ceased. He took away the desire, so that I was no longer sucked in as easily. I learned to flee temptation immediately, get on my knees and pray for help.
This is infinitely easier than doing it on my own. His spirit is able, while I am in submission to Him, to take away the temptation and allow me to see sin as God sees it - wretched and vile, which removes the temptation completely. The load is lightened, and it is easier than struggling on my own. To me, that is what those verses mean.