I have a problem in that I have a lot of interior painting to do and in some cases it is holding up carpentry because I want to prime, for example, baseboards before installing them. However, after priming with an oil-based paint, I need to sand but I can't sand because the oil primer is rubbery and will tear out if I sand it. I have heard that it is supposed to "cure" and then will sand to a powder, but the problem is that it looks like it may take weeks before it dries that much and it is holding up the whole project. How do I deal with this problem?
24 hours is normal
The problem is, people who have only painted with latex paint have funny expectations about drying.
Painting, and then sanding, in the same workday, is out of the question. 8 hours of dry isn't going to cut it for any alkyd, if the next step is to sand or switch from prime to topcoat.
If it's taking more than 8 hours to get tack-free, or if it's taking longer than 24 hours to be sandable, you may be dealing with adverse conditions such as low temperature or poor air circulation. You need to change the air periodically because it will saturate with the paint solvents, and that slows drying. (just like high humidity slows drying of water). If your conditions are good, the next step is to talk to your paint supplier, because some paints are extra slow - they may have quick-dry additives available, to bring them into a "24 hours to sand" expectation.
By the way, "curing" is a red herring. For 1-part paints, curing takes months or even years. If it won't sand, it's not even dry.
Use primer for priming
In my experience, alkyd primers dry much faster than alkyd topcoats, which is why I'm surprised you're having a problem with primer. If you're actually painting a topcoat, and relying on some advertising claim that the paint is its own primer, yeah, that's a lie. That's always a lie. You'll never see a pro painter doing that.
At the risk of stating the obvious, read the instructions
I think that's actually a law: it can do environmental damage, so you must use the product consistent with its instructions or labeling. A careful read of the paint-can fine print is essential, and if the fine print says "See instruction booklet" (as my 2-part paints do, as there isn't enough room on the label for critical warnings and also instructions), then you need to do that. Not reading the instructions is the surest way to have mis-set expectations and a failed application.
Also note that it is likely the instructions will call out certain "re-coat windows" which you have to hit. This is because of the (superior) nature of alkyd paints. You generally have two recoat windows: between X and Y hours (when the paint is dry enough to be stable yet soft enough to chemically bond to the next coat); and beyond Z hours (when chemically bonding is no longer possible, but the paint is nice and hard so it can be scuff-sanded so the next coat can mechanically bond). Adverse conditions can warp those numbers; conditions for slow drying extend them, etc.
Chemical bonding has superior performance to mechanical bonding. If you've ever seen paint layers separate, that is someone relying on a mechanical bond and failing to do proper surface prep by scuff-sanding and cleaning.
You can get a paint dryer, or oil drying agent, also known as siccative, from professional paint suppliers.
(This is related to the process known as "japanning".)
An "oil base" paint will have solvent ( not water) vehicle ; drying depends on which solvent and the temperature and air circulation. But as you note it must "cure" - (polymerize) to develop film strength before you can sand, cure rate also depends on temperature. Increased temperature and air circulation is the most practical way to speed it up. In a worst case situation , a coating will never cure properly at some low temperature.
When we are doing trim on a house we often use oil based paints including primer. It simply lasts longer and looks great. If we are under a crunch they will dry faster with a box fan and a space heater. Either one will help, both make the drying time way faster.
You still need a good 16 hours - so usually if crew primes trim on day 1 and at end of day throws on a box fan and space heater - then we are good to go the next morning. I will also add when sanding the foam sanding pads (sometimes used for sanding mud) are your best bet here. They are less likely to tear the paint because you can sand with less direct pressure.
(Note that the process of adding heat - even at 85-90 degrees - can cause lighter colored oil based paints to "yellow". This is normal and will happen over time no matter what. If you use a heating element to help dry expect a shade or two difference which may be undetectable unless you put the colors right next to each other. Also having a room at 85-90 is fine with airflow. You should not have a space heater right next to a piece of trim making that area 150 degrees.)