When it shuts down do you have spark?
What "New" distributor did you get? Was there a problem with the original GM HEI? Did you hang onto the original?
The Asian HEI distributors have less than stellar internals. I've seen more serious electrical issues with these than they are worth. Most of them do not use the stock pickup coil and reluctor and the "modern" Chinese equivalent is garbage.
If you have the old one you're usually far better off rebuild it. It's not difficult and you don't usually have to tear it all the way down.
Make sure the steel advance pivot pins aren't worn out. This is not the plastic bushings but the actual pins. That's about the only real deal breaker for a DIY. They can be repaired but require a skill set the average DIY doesn't have...
The AC Delco Professional series parts are a little more coin but they work and keep working.
$20 AC Delco D221 Module Pigtail with capacitor
http://www.rockauto.com/catalog/moreinfo.php?pk=532284$34 AC Delco D1906 Professional Ignition Control Module
http://www.rockauto.com/catalog/raframecatalog.php?catalog=301&partnum=D1906$10 Moroso 26140 Distributor gear shim kit (measure end play before you buy)
$15 Moroso 72300 advance weight kit to replace rusty springs and missing pivot bushings.
Base rebuild $54-$79...
$25 Brass terminal cap and rotor set. Unfortunately the Standard Motor Products DR318X & DR450X Blue Streak Brass Terminal Rotors are getting hard to find. The MSD 5501 or the Accel 8123 brass terminal cap and rotor sets can be had for around $25.
You're at $105 now.
$60 ACDelco D504A Professional Ignition Coil HEI. If you need one. Accel coils and the cheap Asian crap have a rep for eating ignition modules.
If you do a Cap Rotor and quality coil you are at $165
$45 ACDelco D1907X Professional Ignition Distributor Pickup in the unlikely event it's failed (very unlikely)
$25 ACCEL 31035 HEI Adjustable Vacuum Advance Canister if the vacuum can has failed. Again very unlikely.
Absolute worst case $235 for a fully rebuilt GM HEI distributor that should last another 40 years.
2-3 hours to clean and re-assemble.