The short version is that yes NPCs do ramp up more quickly then players those last couple of levels before a new ranks. It's difficult to compensate as a newbie. It's not so hard for veteran players.
Jumping to another ship may not be the solution for you. Cruisers are bigger and tougher, but they are not as nimble as Science Vessels. And Escorts are definitely more tactical, but they are also glass cannons (e.e. go fast. go splat.

).
Your new Tier 3 ship will definitely give you a big bump in capabilities. For a Science Vessel, it's another weapon slot in front, a Lt Commander Science slot, Lt Tactical and Engineering stations, more hull strength. You get a signification better tactical and survivability capabilities with the better Bridge Stations. They make a big different.
Note: I didn't like the Golf ball Science Vessels at first, but they eventually grew on me. I got to like the Horizon Class with the Hope Class nacelles.
IF you want a different type of T3 Science Vessel, you may look at the Nebula Class Science Vessel. It's a C-Store purchase, and it's nothing like the Golf ball ships. It's more a cruiser, it turns slower, has more crew and hull, but it has a Universal Ensign Station in place of the Ensign Science Station. You can go with a any BOFF to give you more flexbility instead of just the Science BOFF.