Can a bad battery affect my MBP's performance, even when plugged in?

Question

Well, as the title says... my battery status says "Service Battery", and although I know the instructions are simple ("dude, take your laptop's battery for servicing!"), a few months ago my financial situation changed, I'm trying to adjust the budget as much as I can for now, and I'm not sure I can afford to spend much money at the moment.
The thing is, my laptop's performance has been slowly getting worse all the time. at first, it was only with resource-intensive applications, but lately, even having more than 5 tabs open in Chrome gets the system all laggy. Now, as I've read on other threads talking about batteries being in bad shape, I would assume that, if the system can't draw much power, it can't work as hard, so a bad battery could be the culprit of such performance. However, if my laptop's plugged in, shouldn't it be drawing power straight from the outlet, hence being able to get all the juice it needs to feed both the 9400M & 9600M GT?
Ah! by the way, the info I guess I shouldn't post this without: It's a Macbook Pro 5.3, purchased in March 2010. This is System Profiler's Power section info:
Battery Information:
  Model Information:
  Serial Number:          W0948AB7V77VC
  Manufacturer:          SMP
  Device name:          bq20z451
  Pack Lot Code:          0000
  PCB Lot Code:          0000
  Firmware Version:          0003
  Hardware Revision:          0003
  Cell Revision:          0100
  Charge Information:
  Charge remaining (mAh):          3991
  Fully charged:          Yes
  Charging:          No
  Full charge capacity (mAh):          4091
  Health Information:
  Cycle count:          446
  Condition:          Check Battery
  Battery Installed:          Yes
  Amperage (mA):          -304
  Voltage (mV):          12294
System Power Settings:
  AC Power:
  System Sleep Timer (Minutes):          10
  Disk Sleep Timer (Minutes):          0
  Display Sleep Timer (Minutes):          10
  Automatic Restart On Power Loss:          No
  Wake On AC Change:          No
  Wake On Clamshell Open:          Yes
  Wake On LAN:          Yes
  Current Power Source:          Yes
  Display Sleep Uses Dim:          Yes
  GPUSwitch:          2
  Battery Power:
  System Sleep Timer (Minutes):          10
  Disk Sleep Timer (Minutes):          10
  Display Sleep Timer (Minutes):          2
  Wake On AC Change:          No
  Wake On Clamshell Open:          Yes
  Display Sleep Uses Dim:          Yes
  GPUSwitch:          2
  Reduce Brightness:          Yes
Hardware Configuration:
  UPS Installed:          No
AC Charger Information:
  Connected:          Yes
  ID:          0x0100
  Wattage (W):          85
  Revision:          0x0000
  Family:          0x00ba
  Serial Number:          0x00a30633
  Charging:          No
Power Events:
Next Scheduled Events:
  Type:          Wake or Power On
  Scheduled By:          Repeating Event
  Time:          10/23/13 9:00 AM
Repeating Events:
  Type:          Wake or Power On
  Time:          9:00 AM every day
Hope someone can help... thank you so much in advance!

Answer 1

Not necessarily, if the battery is trying gobble some power, as the bad batt is still in the charge/power circut. 446 cycles is quite a bit, and not a bad run, it also depends on how it has been used.
I would replace it ASAP as it is possible that it may create/cause more severe issues. A new battery is less $$ than more serious damage that could possibly result from leaving it in.