There must be a few hundred threads here about which anchors are best. Usually the Rocna or Manson Supreme (knock off of Rocna) are considered most reliable.
It is really not advisable to leave your boat anchored for days without someone being on board. Weather has a habit of changing.
Slips are quite expensive and you would be lucky to find one for $50/night. A cheaper alternative is to rent a 'transient' mooring somewhere inexpensive. In NYC you can rent a mooring by 79th St. for $30/night (this is also where the anchorage is). Marinas : West 79th Street Boat Basin : New York City Department of Parks & Recreation
The boat club in Nyack (on Hudson) rents transient moorings for $25/night I think.
Moorings are usually quite secure but not foolproof. Mooring pennants can chafe through (same for an anchor rhode) and old chain can decide to part. Many responsible boaters will do an 'anchor watch' through the night especially if the weather is intense. Many GPS units can be set to trigger an alarm if the location changes more then a specified range.
Boating ain't cheap but there are ways to do it cheaper.