When it comes to unemployment and severance it is, in fact, a state-by-state issue. And some states won't let you get unemployment if you are receiving severance. All it means is that you may have to wait X amount of weeks before you are eligible.
And taking a lump sum severance rather than a week-by-week payout doesn't help you any. In states where severance is an issue, they will look at your lump sum and calculate what that works out to in terms of a weekly wage—and then require that you wait that number of weeks until you are eligible.
Now that said, some states don't count it at all—California, for example. According to the unemployment website: "Severance pay is not deducted from unemployment insurance benefits and does not affect your eligibility to receive benefits. The method of payment, such as a lump sum payment or payments paid to you at regular pay period intervals, does not change the nature of the payment. However, you must report severance pay at the time you file your unemployment insurance claim."