The correct router bit for cutting shapes out of plywood

Rough out the cut with a jig saw (or band saw), followed by a router with a small-diameter large-shank bit for a smooth finish.

I think that measurement is needed to determine what the problem is.

Load up the router with a bit, and measure how much the bit protrudes from the collet.

Lock the base in place. Measure the distance from the collet face to the base.

Measure from the base to the router body.

Now run the router until the bit sinks from the work.

Redo all those measurements, this will tell you which part is slipping. eg. If the bit is sticking out less from the collet, then the collet isn't holding the bit -- now you would have to figure that out -- is the lock nut tight, is the collet dirty or damaged.

If the distance from the base to the collet face is different then either the router body has slipped, or the depth adjustment has slipped. Then check the base to router body distance.

I have one old router that I can't trust the depth on - when it's on and upright the bit could choose to stick out 1/32 further than when it's upsidedown -- this router has an issue with the bearings and the shaft will wiggle up and down - this might also be an issue with your router, but it sounds like the travel is too far for this to be the problem.