We don't have things as perfect as you do, so my advice won't be great, but we get paid every other week.
At the moment, we get paid on opposite weeks, but my check is for quite a bit more than DHs, and it's a predictable number, wheras
dh's can fluctuate by hundreds of dollars. What I do is pay my bills online. Even
if the bill isn't due until say, the 30th, I have it set up to get paid slightly earlier and deduct it from our pay right then. It works out that I have a bunch of bills that come out on the 9th and a bunch that come out on the 18th. This way it doesn't appear that I have "all this money" sitting in the bank when really I dont (which you will know anyway with the budget.)
I'd say you probably need a longer budget, not a shorter one. Probably a 6 week or 2 month budget, since you are used to getting paid on certain dates, not certain DAYS of the month. I think it will be confusing at first but no more difficult than before.
You can totally do it!