If a climb is steep enough, I would aggressively lean forward and tension my arms. Not quite getting out of the saddle, but just-about.
This shifts more weight to the front wheel, which for me gets a little "lifty" on the steep climbs. The steepest gradient is therefore limited to an angle where I can keep the front wheel on the road while maintaining some forward speed above 4 km/h. Any less and I am doing more of a long-trackstand and spend too much effort on balancing instead of going forward, eventually having to put a foot down and then it's always too steep to start.
As for sliding around on the saddle, this could be tweaked in two ways.
Adjust your saddle's angle so the nose is down a little. If your saddle is a single-bolt design you might have to move it a whole tooth, or if its a two-bolt design then just a couple of turns looser on the rear then the same tighter on the front bolt.
Is your saddle too narrow? As we age, our ability to conform to the bike decreases. It may be that a saddle with a wider sit-bone area may reduce your rearward slide tendancy.
Saddle position is a compromise between climbing and flat-land (downhill is less-important) so mind that any adjustments to help one don't hinder the other.