There are several ways you can pronounce this contraction, to make the sound more relaxed.
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A quick study in the pronunciation of the contraction ‘we will’. ‘We’ has the ee as in she vowel. However, when we contract it, we rarely pronounce the sharp ee sound. We will, if we want, for emphasis: ‘We’ll do it tomorrow!’ But most of the time it takes on a different vowel sound. And that is, the ‘uh’ as in ‘pull’, or the ‘ih’ as in ‘sit’. And they are pretty much interchangeable. Uh as in pull: ‘W[uh]’ll do it when we feel like it’: w[uh]’ll, w[uh]’ll. you see the sides of the mouth come in just a little bit and then this part of the mouth comes out just a little bit: ‘W[uh]’ll do it soon.’ … ‘W[uh]’ll do it tomorrow.’ Now, as I said, that is interchangeable, that uh, with an ih as in sit: ‘W[ih]’ll do it soon.’ … ‘W[ih]’ll do it tomorrow.’ Those were the ih, ih. Either one of those is correct in normal conversation. As I said if you want to emphasize, then you can use the ee as in she: ‘We’ll do it!’
On a related note, the word ‘will’ is pronounced with this ih as in sit. But it can also be exchanged for the uh as in pull: I will do it soon, I w[uh]ll do it soon. The first one with with an ih and the second one was with an uh. And now in conversation, when one is speaking quickly, these two sounds don’t have a lot of distinction and either one is fine in the pronunciation of the word ‘will’.