Yes, the two parks FP computer systems are not connected. So you can grab a FP for a DL attraction and then head right over to DCA and get one immediately.
"Hold" a FP is a bit of a confusing term. There is no limit to how many FPs you can "hold" at any one time, even if you stay at one park.
If you mean "obtain" FP, then yes there are limits to how often you can obtain them, and yes, DM is correct that obtaining FPs at DL does not block or delay you from obtaining them at DCA. And vice versa.