On the 35 Rem, I took a deep breath, centered the reticles, and started over. I got it sighted in OK, and shot some 1 1/2" 3 shot groups at 50 yds. OK so far ! All my other barrels have the same Bushnell scope on them, and "high" rings have worked fine. On the 35 I have a Bushnell Banner, and it really needed about 1/16" more clearance over the hammer. I looked hard and found Weaver extra-high Sure Grip rings that give 0.506" clearance. They are steel, and have 2 screws on each side. The original Weaver rings with screws on only one side made me crazy; getting the reticle leveled was very hard, since the screws always rotated the scope a little. Anyway, I just bore sighted it with the extra-high rings and fired a confirming shot, and I'm good to go.
On the 243, I finally decided to send the scope to Bushnell to make sure the adjustments were doing what they should, and to replace the cheap little plastic finger knob on the elevation adjustment. For the last 4 days, their repair status page has said that they are preparing to ship. Hmmm....right ! I recently sent them some binos to fix, and they were a lot faster. Anyway, a lifetime warranty is good, so I'm just waiting patiently...
Thanks for the helpful comments, guys !