I have some gaps between my subfloor and joists. Some are causing squeaks and creaks. Should I fill in the gaps with construction adhesive or screw the subfloor to joist or do both? Thanks!

Screws hold well. You may want to see if the surface is flat before you pull the sub-floor down to the joists. If the floor is flat, now, you can add a sister joist next to the original underneath and glue the new joist to both the existing joist and the bottom of the sub-floor presuming that the area below the joists is accessible.. If you have a hump in the joist and need to flatten the floor by grinding with an edger and coarse paper or plane it with an electric plane, the screws will need to be counter sunk so pre-drilling is necessary.
When using screws to fasten the sub-floor tight, you will sometimes need to use a pilot hole just to get the screw to pull the sub-floor down. The hole through the sub-floor will need to be the size of the threads, not the thinnest part of the shank, if the threads go all the way to the head. Some screws have a straight shank with threads only on the part that go into the joist, so it will pull the sub-floor down completely to the joist without a pre-drill.

SPAX screws are a good choice for this. HD sells them. If the subfloor is nailed and the movement is around the nails then removing the nails and replacing them with screws will add additional hold.

Thanks for the replies. I’ve should have added that the hardwood is already down. I’m working from below in the unfinished basement. I was hoping just some construction adhesive might help with the small gaps?

In that case get some wood shims and insert them between the subfloor and joist. Just slide them in hand tight. Don't hammer the shims or you may cause the subfloor to lift in another area. You can spread some wood glue on the shims if you would like. You can cut the protruding end of the shim or just leave them long. When you install the shims make sure nobody is standing over that area......you just want to immobilize the subfloor to keep it from moving when somebody does walk over the area.