You live in a relatively warm, humid environment so I definitely would not recommend poly sheeting, especially on the interior surface. That's a great way to get condensation and wood rot. You need to have a way for the vapor to escape and the structure to dry out, which airtight plastic doesn't provide.
You don't mention what type of blown insulation you have now: fiberglass, cellulose, or mineral wool. The simplest and most cost effective way to deal with it is to just blow more insulation on top of what you have (Personally, I greatly prefer blown cellulose or rock wool batts over any fiberglass.) You can easily rent a blower from any big-box hardware store and blow some insulation in a weekend.
Like @Isherwood mentioned, make sure any recessed lights are IC rated or put barriers around them to keep the insulation at a safe distance. Even better, replace them with airtight led fixtures.
The next step up is to add soffit and ridge or deck vents, closing off the gable vents, so cool air comes in as low as possible and exits as high as possible. This will help drop your attic temp during the summer cooling season.
The next level of improvement, as you considered, is removing the existing insulation, air sealing your ceiling, and re-insulating. I'd suggest either the house wrap or drywall, not both. Using both would be redundant. Overall, I think the drywall would be easiest to install and seal.
If you want to go all-in, you can add roof vents, then put solid foam sheet insulation on the bottom of your roof deck to make your attic a conditioned space for storage or an extra room.
FWIW, I just removed all the insulation (R-3 to R-6) in my attic, air sealed, and am re-insulating with rock wool batts. (I would do blown cellulose, but my lady didn't like the dust factor when blowing.) Just the air sealing alone lowered my heating bill this past winter, with only half the house insulated to R-15. I expect even better when I finish insulating the ceiling to R-45. If you want to know more about the insulation removal, just post a separate question.