Whenever somebody says "I'll do as best as I can", it always sounds like "I'll do asbestos I can." Obviously I know what they meant, but it's not what my brain heard.
Enunciation matters. If you are speaking, be clear. Take your time; it always takes less time to be clear than it does to be confusing and then have to clarify.
For the above example, put a half- or quarter-beat between "best" and "as". It will not only make sure that you don't mash your words together, but it will also emphasize "best" very subtly.
Practice saying "as best as I can" out loud a little bit. Try it with different amounts of space between the words and see how it affects the tone of the phrase.