How to avoid writing a grant with an advisor
Short answer #1: Talk to your adviser about these things. Tell him that you wonder how joint proposals will be perceived and get his perspective on things.

Short answer #2: Remember that it's easy to write proposals, and hard to write winning proposals. How are you going to learn what to say and not say in a proposal? The easiest way to learn this skill is to work with someone who has done this many times and who may have been on review panels many times. I have reviewed many a proposal by junior scientists in my lifetime where it is clear that they mean well, but have essentially no idea what to write in a proposal that can be funded. That may be your fate if you write by yourself, unless you have an experienced mentor who is willing to help you with it.

As far as demonstrating independence: If you're already on the job market, then this proposal will not be funded before anyone will look at your file anyway, so it's not going to make a difference one way or the other. It might make a difference come time for tenure, but that is so far in the future that you will have plenty of opportunities to demonstrate your independence till then. And for the moment, any funded proposal is better than no funded proposal.