Can’t open console in Oblivion

For the save game bloating (which could be the reason for the crashes), you might want to try this mod:
Clean up by kuertee

Description
Prevents saved-game file-size bloating by cleaning the game of actor and item references that are no longer required. (e.g. inactive references created from dropped items that were picked-up by the player.)

Cleans long-dead actors that may have been corrupted somehow.

Details
The cleaning process used is the game's inherent behaviour of removing items that no longer has any "links" to the game world.

Normally, dead actors and "dynamically created" items that were not picked up by the player will be removed from the world, when the cell they are in resets. (Dynamic items are those created at game-time as opposed to those created in the Construction Set. Examples of these items are: those created by PlaceAtMe, those dropped by the Player, those dropped by dying NPCs: e.g. shields, etc.)

However, there are cases when these actors and dynamically created items will (for some reason or another) stay in the game world forever. And actors not cleaned up by the game properly will never respawn, causing a "broken" game experience. Saved-game bloat is the result of these. And saved-game bloating may be the cause of some game crashes.

To prevent crashes, I'd recommend Streamline's Streampurge feature (memory purging on an intelligent, scheduled basis - can be configured to do a memory purge before a save, which can help with stability; more info here) and Fast Exit 2 by Scanti. You can choose to enable only Streampurge and Streamsave and not the other Streamline features in sl.ini (Streamline's configuration file).

If you are running a 64-bit OS and have 4 GB or more RAM, you might want to try the 4GB EXE Patcher on Oblivion.exe . More info on the Steam forums thread on this link here.

You could also try OSR (Oblivion Stutter Remover):

This plugin makes Oblivion not "stutter" as much, and generally feel smoother or perform better. It prevents or mitigates a number of issues related to stuttering and framerates, and can reduce the frequency of stutter related crashes.

Make sure you set MaximumFPS = 0 in 'sr_Oblivion_Stutter_Remover.ini' to avoid it putting an artificial cap in your framerate. Aside from that, try using it with the default options. If it still crashes, you might want to try to replace the default Oblivion heap (memory manager) with one of OSR's. In 'sr_Oblivion_Stutter_Remover.ini', set bReplaceHeap = 1 and change iHeapAlgorithm to 1, 2 or 5 - read 'sr_Oblivion_Stutter_Remover.txt' for more info and other OSR tweaks that you could try.

Note that Streamline, Clean-up, Fast Exit 2 and OSR requires OBSE (Oblivion Script Extender)

After installing any new mods, make sure to run BOSS (Better Oblivion Sorting Software). BOSS properly sorts the load order of the mods, avoiding crashes and mod conflicts. It also provides a large number of usage notes for many mods, including requirements, incompatibilities, updates and bug warnings.

More info on configuration and mods that can help stabilize and prevent crashes in Oblivion here: http://tescosi.com/wiki/Oblivion/Stabilization_and_Optimization

Make sure you have set bAllowConsole to 1 in your oblivion.ini file.

It's likely you had this set on your old PC and not on the new one.